UNITS & DIMESNIONS

DISTANCE DISPLACEMENT

The complete length of the path between any two points Displacement is the direct length between any two
is called distance points when measured along the minimum path
between them
Distance is a scalar quantity as it only depends upon the Displacement is a vector quantity as it depends upon
magnitude and not the direction both magnitude and direction

Distance can only have positive values Displacement can be positive, negative and even
zero

Distance travelled is measured over the trajectory Displacement depends only on the initial and final
position of the body and which is independent of
the trajectory

Its value matches the magnitude of the displacement Its magnitude coincides with the distance traveled
vector when the trajectory is a straight line and there is when the trajectory is a straight line and there is no
no change of direction. change of direction.

Its magnitude always increases when the body is in Its magnitude increases or decreases with the motion
motion, regardless of the trajectory according to the trajectory described

SI Unit: meter SI Unit: meter

Distance = Speed * Time Displacement = Velocity * Time

Distance can never decrease with time Displacement can decrease with time

It is not the unique path It is the unique path between two end points

DISTANCE ≥ DISPLACEMENT

EXERCISE - 1

1. An object travels a distance of 5m towards east, then 4m towards north & then 2m towards west.
Calculate the total distance travelled and total displacement?
2. A man has to go 50 m due North, 40 m due east and 20 m due south to reach a field.
a) What distance he has to walk to reach the field?
b) What distance he has to walk to reach the field?
c) What is his displacement from his house to the field?

1
UNITS & DIMESNIONS

3. A body is moving in a straight line. Its distance from origin are shown with time in fig A, B, C, D
and E represent different parts of its motion. Find the following:
a) Displacement of the body in first two seconds
b) Total distance travelled in 7 seconds
c) Displacement in 7 seconds.

4. A particle starts from the origin; goes along the X-axis to the point (20m,0) and then returns along
the same line to the point (-20 m, 0). FGind the distance and displacement of the particle during
the trip?
5. A wheel of radius 1 m rolls forward half a revolution on a horizontal ground. The magnitude of
the displacement of the point O the wheel initially in contact with the ground is:
a) 2 π
b) √2 π
c) √𝜋 ∗ 𝜋 + 4
d) 𝜋
6. An object moves 10 m towards East then 10 m towards north and from that point it moves 10 m
vertically upwards. Find the distance and displacement of the object.
7. An object moves along the grid through the points A, B, C, D, E, and F as shown below.
a) Find the distance covered by the moving object.
b) Find the magnitude of the displacement of the object.

8. A farmer moves along the boundary of the square field of side 10m in 40 seconds. What will be
the magnitude of displacement of the famer at the end of 2 minute 20 seconds from its initial
position.
9. The minutes hand of wall clock is 10cm long. Find its displacement and distance covered 10 am
to 10.14 am
10. Find the distance and displacement when an object moves in different positions:

2
UNITS & DIMESNIONS

SPEED VELOCITY
It is the distance travelled by a body per unit time in any It is the displacement of a body per unit time in a
direction particular direction
Scalar Quantity Vector Quantity
Speed may be positive or Zero but never negative Velocity may be Positive, Negative or Zero
It is the rate of change of distance It is the rate of change of displacement
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
Speed = 𝑡𝑖𝑚𝑒 Velocity = 𝑡𝑖𝑚𝑒
SI Unit: m/s SI Unit: m/s
Speed can never decrease with time Velocity can decrease with time
Its magnitude always increases when the body is in Its magnitude increases or decreases with the motion
motion, regardless of the trajectory according to the trajectory described
It gives an idea about rapidity of motion of body It gives an idea about rapidity as well as position of
body in motion
SPEED ≥ VELOCITY

AVERAGE SPEED AVERAGE VELOCITY

It is the total distance travelled by a body per unit It is the total displacement of a body per unit time
time in any direction in a particular direction
Scalar Quantity Vector Quantity
Speed may be positive or Zero but never negative Velocity may be Positive, Negative or Zero

It is the rate of change of total distance travelled It is the rate of change of total displacement
covered
𝑇𝑜𝑡𝑎𝑙 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑙𝑒𝑑 𝑁𝑒𝑡 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
Average Speed < s > = Average Velocity <v> = 𝑇𝑜𝑡𝑎𝑙 𝑇𝐼𝑚𝑒 𝑇𝑎𝑘𝑒𝑛
𝑇𝑜𝑡𝑎𝑙 𝑇𝑖𝑚𝑒 𝑇𝑎𝑘𝑒𝑛

SI Unit: m/s SI Unit: m/s

Average Speed can never decrease with time Average Velocity can decrease with time
AVERAGE SPEED ≥ AVERGAE VELOCITY

➢ The slope of the chord (secant)

provides the average Velocity
Instantaneous Velocity & Instantaneous Speed
➢ The quantity that tells us how fast an object is moving
anywhere along its path is the instantaneous velocity.
➢ It is the average velocity between two points on the path
in the limit that the time (and therefore the displacement)
between the two points approaches zero.

➢ The magnitude of instantaneous velocity is always equal

to that of instantaneous speed.

3
UNITS & DIMESNIONS

ACCELERATION RETARDATION
It is the rate of change of velocity It is the rate of change of velocity
Vector Quantity Vector Quantity
Sign Convention used for this is Positive Sign Convention used for this is Negative
It is the rate of change of total distance travelled It is the rate of change of total displacement
covered
𝑉2 − 𝑉1 𝑉− 𝑉
Acceleration = Retardation= 𝑡2 − 𝑡 1
𝑡2 − 𝑡1 2 1
SI Unit: m/s2 SI Unit: m/s2
Direction of rate of change of velocity and Direction of rate of change of velocity and velocity
velocity are same, body is said to be are opposite, body is said to be retarded.
accelerated.
Velocity keeps increasing. Increased velocity starts decreasing.
Initial Velocity < Final Velocity Initial Velocity > Final Velocity
𝑣⃗ .⃗⃗⃗⃗
𝑎 > 0 (Dot Product of velocity & acceleration 𝑣⃗ .⃗⃗⃗𝑎⃗ < 0 (Dot Product of velocity & acceleration is
is greater than zero) less than zero)
The angle between velocity and acceleration is The angle between velocity and acceleration is
acute (< 90°) obtuse (>90°)

AVERAGE ACCELERATION
Average acceleration between two points P1 and P2 is defined as the ratio of the variation of the
velocity and the time used to complete the motion between both points:
∆𝑣 𝑣2 − 𝑣1
a average = ∆𝑡 = 𝑡2 − 𝑡1

Note:
➢ If anybody is accelerated a1 till time t1 and a2 up to next time t2 then average
𝑎⃗⃗1 𝑡1 + 𝑎⃗⃗2 𝑡2
acceleration is 𝑎𝑎𝑣𝑔 =
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑡1 + 𝑡2

INSTANTANEOUS ACCELERATION
➢ Acceleration at any instant of time in velocity time graph

➢ The slope of v-t graph at any instant of time provides

Instantaneous acceleration

4
UNITS & DIMESNIONS

3. Equations of Motion – Uniform Accelerated Motion

𝒗 = 𝒖 + 𝒂𝒕 𝒗𝟐 = 𝒖𝟐 + 𝟐𝒂𝒙 𝟏 𝟐
𝒙 = 𝒖𝒕 + 𝒂𝒕
𝟐

a=
𝒅𝒗
→ 𝒅𝒗 = 𝒂 𝒅𝒕 a=
𝑣𝑑𝑣 Putting 𝑣 = 𝑢 + 𝑎𝑡, in 𝑣 2 − 𝑢2 = 2𝑎𝑥
𝒅𝒕 𝑑𝑥
𝒗 𝒕 𝑣 𝑠 (𝑢 + 𝑎𝑡)2 − 𝑢2 = 2𝑎𝑥
∫ 𝒅𝒗 = ∫ 𝒂 𝒅𝒕 ∫ 𝑣𝑑𝑣 = ∫ 𝑎𝑑𝑥 𝑢2 + 𝑎2 𝑡 2 + 2𝑢𝑎𝑡 − 𝑢2 = 2𝑎𝑥
𝒖 𝒐 𝑢 0 1
𝒗 – 𝒖 = 𝒂𝒕 𝑣 2 − 𝑢2 = 2 𝑎 𝑥 𝑥 = 𝑢𝑡 + 𝑎𝑡 2
2
𝒗 = 𝒖 + 𝒂𝒕 𝑣 2 = 𝑢2 + 2𝑎𝑥

➢ Distance Travelled in nth second

If Sn is the distance travelled by an object in the nth second & Sn-1 is the distance travelled in
(n-1)th second then:-
𝟏
Sn = Sn – Sn-1 = un + ½ an2 – [ u(n-1) +𝟐a(n-1)2]
𝒂
Sn = u – 𝟐(1- 2n)

𝒂
Sn = u + 𝟐 (2n -1)
➢ Displacement in terms of initial velocity and final velocity
v2 = u2 + 2 a x
v2 - u2 = 2ax

(v + u) (v - u) = 2 a x

(𝒗+𝒖) (𝒗−𝒖)
=x
𝟐 𝒂

𝒖+𝒗
x=( )𝒕
𝟐

➢ Average Velocity
𝒖+𝒗
<vavg> = 𝟐
𝒖+𝒗
Since 𝜟𝒙 = ( )𝒕
𝟐
𝜟𝒙 𝒖+𝒗
= ( )
𝒕 𝟐
𝒖+𝒗
<vavg> = ( )
𝟐

Difference Between Distance and Displacement – only occur when u and a change
Case1: When u is 0 or u and a are parallel to each other (Ɵ = 0° or < 90°) → motion is simply
accelerated and, in this case, distance is equal to displacement.

Case2: When u is antiparallel to a (Ɵ > 90°) → in this case, distance is not equal to
displacement.

5
UNITS & DIMESNIONS

➢ 3 Equation of Motion in Vector form

⃗⃗ = 𝒖
a) 𝒗 ⃗⃗ + 𝒂
⃗⃗𝒕
b) (𝒗
⃗⃗. ⃗𝒗⃗) = (𝒖
⃗⃗. 𝒖
⃗⃗) + 𝟐(𝒂
⃗⃗. 𝒙
⃗⃗)
𝟏
⃗⃗ = 𝒖
c) 𝒙 ⃗⃗. 𝒕 + 𝒂⃗⃗. 𝒕𝟐
𝟐

➢ Symbols: ➢ Conditions:

a) Initial Velocity → u a) When an object starts from rest → u = 0

b) Final Velocity → v b) When an object comes to rest → v = 0
c) Uniform Acceleration → a c) Object accelerating → a = +ve
d) Time – t d) Object de-accelerating → a = - ve
e) x – Displacement e) X in +x direction → x = + ve
f) X in -x direction → x = - ve
Note
➢ All the above equations are only valid for Uniform Accelerated Motion

UNIFORM & NON UNIFORM

Uniform Speed ➢ An object is said to be moving with uniform speed, if it covers equal
x distances in equal intervals of time, howsoever small these intervals
may be.
➢ Uniform speed is shown by straight line in x-t graph
➢ Slope of the curve Ɵ = 45°
➢ Speed is Constant
t ➢ Acceleration is zero (Linear Motion)

Non-Uniform Speed ➢ An object is moving with non-uniform speed if it covers equal

x distances in unequal interval of time or unequal distances in equal
interval of time, howsoever small these intervals may be.
➢ Speed is Variable – Increasing and decreasing
➢ Acceleration – accelerating and retarding.

Uniform Velocity ➢ An object is said to be moving with uniform velocity, if it covers equal
displacement in equal intervals of time, howsoever small these
x
intervals may be.
➢ Uniform Velocity is shown by straight line in x-t graph.
➢ The magnitude and direction of the velocity of the body remains
same at all points.
t ➢ Velocity is constant
➢ Acceleration is zero

6
UNITS & DIMESNIONS

Non-Uniform Velocity ➢ An object is moving with non-uniform velocity if it covers equal

distances in unequal interval of time or unequal distances in equal
x interval of time, howsoever small these intervals may be.
➢ Velocity does not remain constant

Uniform Motion ➢ If the velocity of the particle remains constant with time, it is called
uniform motion or motion with uniform velocity.
x ➢ The x-t graph is a straight line.
➢ Acceleration is zero.
➢ Magnitude of velocity is constant and Direction of velocity is fixed
➢ Straight line motion
➢ 1 D motion
t

Non-Uniform Motion ➢ If the velocity of the particle changes with time it is called
accelerated or Non-Uniform Motion

Uniform Acceleration ➢ An object is said to be moving with uniform acceleration if its velocity
changes by equal amounts in equal interval of time
v ➢ The velocity of the body which is moving in straight line changes at
a constant rate
Example:
1. a) Motion of a ball rolling down on an inclined plane.
t 2. b) Motion free falling body from certain height
3. c) A car moving with uniform velocity in a straight line

Non-Uniform ➢ An object is said to be moving with non-uniform acceleration if its

Acceleration velocity changes by un-equal amounts in equal interval of time.
v

7
UNITS & DIMESNIONS

EXERCISE - 2

1. A car is moving along a straight-line OP. It

moves from O to P in 18 seconds and returns
from P to Q in 6.0 seconds. What is the
average velocity and average speed of the car
in going a) from O to P? and from b) O to P
and back to Q.
8. On an open ground a motorist follows a track
that turns to his left by an angle 60° after every
500m.Starting from a given turn specify the
2. A table clock has its minute hand 4.0 cm long. displacement of the motorist at the third, sixth
Find the average velocity of the tip of the and eighth turn. Compare the magnitude of
minute hand between 6:00 am to 6:30 am and displacement with the total path length covered
between 6:00 am to 6:30 pm. by the motorist in each case?
3. A man walks at a speed of 10 km/hr for 10 Km 9. A drunkard walking in a narrow lane takes 5
and 20km/hr for the next 20 Km. What is his steps forward and 3 steps backward, followed
average speed for the walk of 2 Km? again by 5 steps forward and 3 steps
backward, and so on. Each step is 1m long and
4. An electron revolves around the nucleus. It
requires 1s. Plot the x–t graph of his motion.
revolves 10 rounds around the nucleus in the
Determine graphically or otherwise how long
first orbit off Hydrogen. Use Bohr theory for
the drunkard takes to fall in a pit 9m away from
radius of the orbit. What is the average speed
and velocity of the electron in the orbit? the start.
10. A man walks on a straight road from his home
5. A person travelling on a straight line moves
with a uniform velocity V1 for some time and to a market 2.5 km away with a speed of 5
with uniform velocity V2 for the next equal time. km/h. On reaching the market he instantly turns
The average velocity is given by: and walks back with a speed of 7.5 km/h. What
𝑉 +𝑉 is the
a) 𝑉 = 1 2
2 (a) magnitude of average velocity and
b) 𝑉 = √𝑉1 𝑋 𝑉2 (b) average speed of the man, over the interval
2 1 1 of time
c) 𝑉
= 𝑉1
+𝑉
1 1
2
1
(i) 0 to 30 min.
d) 𝑉
= 𝑉1
+ 𝑉2 (ii) 0 to 50 min
(iii) 0 to 40 min.
6. A person travelling on a straight line moves 11. A driver takes 0.20 s to apply the brakes after
with a uniform velocity V1 for some distance x he sees a need for it. This is called the reaction
and with uniform velocity V2 for the next equal time of the driver. If he is driving a car at a
distance. The average velocity is given by: speed of 54 km/h and the brakes cause a
𝑉 +𝑉
a) 𝑉 = 1 2 2 deceleration of 6.0 m/s2, find the distance
b) 𝑉 = √𝑉1 𝑋 𝑉2 travelled by the car after he sees the need to
2 1 1 put the brakes on?
c) 𝑉
= 𝑉1
+𝑉
2
1 1 1 12. A passenger is standing d distance away from
d) = +
𝑉 𝑉1 𝑉2 a bus. The bus begins to move with constant
acceleration, to catch the bus, the passenger
7. A particle goes along a quadrant from A to B of runs at a constant speed u towards the bus.
a circle of radius 10m as shown in figure. Find What must be the minimum speed of the
the direction and magnitude of displacement passenger so that he may catch the bus?
and distance along the path AB?

8
UNITS & DIMESNIONS

13. A car accelerates from rest at a constant rate α (b) Position x as function of time t.
for some time, after which it decelerates at a (c) Find the maximum distance it can go away
constant rate β, to come to rest. If the total time from the origin.
elapsed is t. Evaluate
19. Acceleration of particle moving along the x-axis
(a) the maximum velocity attained
varies according to the law a = –2v, where a is
(b) the total distance travelled.
in m/s2 and v is in m/s. At the instant t = 0, the
14. Some information's are given for a body particle passes the origin with a velocity of 2
moving in a straight line. The body starts its m/s moving in the positive x-direction.
motion at t=0. (a) Find its velocity v as function of time t.
Information I: The velocity of a body at the end (b) Find its position x as function of time t.
of 4s is 16 m/s (c) Find its velocity v as function of its position
Information II: The velocity of a body at the end coordinates.
of 12s is 48 m/s (d) Find the maximum distance it can go away
Information III: The velocity of a body at the end from the origin.
of 22s is 88 m/s (e) Will it reach the above-mentioned maximum
The body is certainly moving with: distance?
(a) Uniform velocity
20. The velocity of the particle moving in the + x
(b) Uniform speed
direction varies as v = α √𝑥 where alpha is
(c) Uniform acceleration
positive constant. Assuming that at moment
(d) Data insufficient for generalization
t=0, the particle was located at the point x =0.
15. Position vector r of a particle varies with time t Find:
1 4 a) the time dependence of the velocity and the
according to the law 𝑡2 i - 𝑡1.5 j + 2t k, where
2 3
acceleration of the particle
r is in meters and t is in seconds. Find
b) the mean velocity of the particle averaged
a) Suitable expression for its velocity and
over the time that the particle takes to cover
acceleration as function of time.
first s meter of the path.
b) Magnitude of its displacement and distance
traveled in the time interval t = 0 to t= 4 s. 21. A car starts from rest and accelerates uniformly
16. A particle moving with uniform acceleration for 10 s to a velocity of 8 m/s. It then run at a
passes the point x = 2 m with velocity 20 m/s at constant velocity and is finally brought to rest
the instant t = 0. Sometime latter it is observed in 64 m with a constant retardation. The total
at the point x = 32 m moving with velocity 10 distance covered by the car is 584 m. Find the
m/s. Find value of acceleration, retardation and total time
(a) Acceleration? taken.
(b) Position and velocity at the instant t = 8 s. 22. The position of a particle is given by the
(c) What is the distance traveled during the equation x (t) = 3 𝑡 3 . Find the instantaneous
interval t = 0 to 8 s? velocity at instants t = 2s, 4s using the definition
17. A car accelerates from rest at a constant rate α of instantaneous velocity.
for some time, after which it decelerates at a 23. A particle is moving along X-axis, its position
constant rate β, to come to rest. If the total time 2
elapsed is to evaluate (a) the maximum varying with time as x (t) = 2 𝑡 3 − 3 𝑡 + 1
velocity attained and (b) the total distance (a) At what time instants, is its velocity zero.
travelled. (b) What is the velocity when it passes through
origin?
18. Acceleration of a particle moving along the x-
axis is defined by the law a = - 4x, where a is 24. A particle is travelling along X-axis with an
in m/s2 and x is in meters. At the instant t = 0, acceleration which varies as: a (x) = - 4x
the particle passes the origin with a velocity of (i) Derive the expression for v (x). Assume that
2 m/s moving in the positive x-direction. Find the particle starts from rest at x = 1m.
(a) Velocity v as function of its position (ii) Hence find the maximum possible speed of
coordinates. the particle.

9
UNITS & DIMESNIONS

25. A particle of mass m is projected in a resisting respectively. Initially A is 10 m behind B. What

medium whose resistive force is F = k v and the is the minimum distance between them?
initial velocity is V0.
32. The acceleration of the particle moving in a
(a) Find the expression for position and velocity
straight line varies with displacement as
in terms of time.
a = 2s +1 velocity of the particle is zero at zero
(b) Find the time after which the velocity
displacement. Find the corresponding velocity
becomes Vo/2.
and displacement equation
26. The position of a particle is given by the
33. A train travelling at 20 Km/hr is approaching a
equation x (t) = 3 𝑡 3 . Find the instantaneous
platform. A bird is sitting on a pole of the
velocity at instants t = 2s, 4s using the definition
platform. When the train is at distance of 2Km
of instantaneous velocity.
from pole, brakes are applied which produce a
27. A helicopter takes off along the vertical with an uniform deacceleration in it. At that instant the
acceleration a = 3 m/s2 and zero initial velocity. bird flies towards the train at 60 Km/hr and after
After a certain time t1, a bullet is fired from the touching the nearest point on the train flies
helicopter. At the point of take- off on ground, back to the pole and then flies towards the train
the sound of the shot is heard at a time t2 = 30 and continue repeating itself. Calculate how
s after the take - off of helicopter. Find the much distance does the bird travel before the
velocity of the helicopter at the moment when train stops?
the bullet is fired assuming that the velocity of
34. Two trains one travelling at 15 m/s and at 20
sound is c = 320 m/s.
m/s are heading towards one another along a
28. A train travels from rest at one station to rest at straight track. Both the drivers apply brake
another in the same straight-line distant l. It simultaneously when they are 500 m apart. If
moves over the first part of the distance with an each train has a retardation of 1 m/s2, the
acceleration of f1 m/s2 and for the remainder separation after they stop is
with retardation of f2 m/s2. Find time taken to
35. What is the difference between distance and
complete the journey.
displacement when?
29. The driver of a car moving at 30 m/s suddenly a) u is either || to a or is 0 & the motion is
sees a truck that is moving in the same simply accelerated
direction at 10 m/s and is 60 m ahead. The b) When u is anti-parallel to a & the motion
maximum deceleration of the car is 5 m/s2. is first retarded and then accelerated
(a) Will the collision occur if the driver’s
reaction time is zero? If so, when? 36. A man is d distance behind the bus when the
(b) If the car driver’s reaction time of 0.5 s is bus starts accelerating from rest with an
included, what is the minimum magnitude of acceleration ao. With what minimum constant
deceleration required to avoid the collision? velocity should the man start running to catch
the bus
30. A particle moves along a horizontal path, such
that its velocity is given by v = 3𝑡 2 - 6t m/ s, 37. For motion of an object along the x-axis, the
where t is the time in seconds. If it is initially velocity v-depends on the displacement x as v
located at the origin O, determine the distance = 3𝑥 2 – 2x, then what is the acceleration at x =
travelled by the particle in time interval from t = 2m?
0 to t = 3.5s and the particle’s average velocity 38. A police party is chasing a dacoit in a jeep
and average speed during the same time which is moving at a constant speed v. The
interval. dacoit is on a motorcycle. When he is at a
distance of x from the jeep, he accelerates from
31. Two particles A and B start moving
rest at a constant rate α? Which of the following
simultaneously along the line joining them in
relations is true if the police is able to catch the
the same direction with accelerations of 1 m/s2
dacoit?
and 2 m/s2 and speeds 3 m/s and 1 m/s

10
UNITS & DIMESNIONS

39. A moving car possess average velocities of 47. A particle is moving along the x-axis whose
5m/s; 10 m/s; & 15 m/s in the first, second and 𝑡3
position is given by x = 4 – 9t + 3
. Mark the
third seconds respectively. What is the total
correct statement in relation to its motion
distance covered by the car in 3 sec?
a) direction of motion is not changing at any of
the instant
40. The average velocity of a body moving with
b) direction of motion is changing at t = 3s
uniform acceleration after travelling a distance
c) for 0 < t < 3s, the particle is slowing down
of 3.06 m is 0.34 m/s. If the change in velocity
d) for 0 < t < 3s, the particle is speeding up
of the body is 0.18 m/s during this time, its
uniform acceleration is 48. A particle of mass m moves on the x –axis as
follows: it starts from rest at t =0 from the point
41. A point moves in a straight line so that its
x =0 and comes to rest at t=1 at the point x =1.
displacement x metre time t second is given by
No other information is available about its
𝑥 2 = 1 + 𝑡 2 . Its acceleration is
motion at intermediate times (0 < t < 1). If a
42. A 2 m wide truck moving with a uniform speed denotes instantaneous acceleration of the
Vo = 8 m/s along a straight horizontal road. A particle then
pedestrian starts to cross the road with a a) a cannot remain positive for all t in the
uniform speed v when the truck is 4 m away interval 0 ≤ t ≤1
from him. The minimum value of v so that he b) |a| cannot exceed 2 at any point in its path
can cross the road safely is c) |a| must be ≥ 4 at some point or points in its
path
43. Which of the following statement is correct?
d) a must change sign during the motion but no
a) If the velocity of the body changes it must
other assertion can be made with given
have some acceleration
information
b) If the speed of the body changes, it must
have some acceleration 49. An athlete starts running along a circular track
c) if the body has acceleration, its speed must of 50 m radius at a speed 5 m/s in the clockwise
change direction for 40 s. Then the athlete reverses
d) if the body has acceleration, its speed may direction and runs in the anticlockwise direction
change at 3 m/s for 100 s. At the end, how far around
𝑑𝑣 𝑑 |𝑣| the track is the runner from the starting point?
44. What does | | 𝑎𝑛𝑑 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡
𝑑𝑡 𝑑𝑡
50. Car B is travelling a distance d ahead of car A.
a) Can these be equal
𝑑 |𝑣| 𝑑𝑣 Both cars are travelling at 60m/s when the
b) Can 𝑑𝑡
= 0, while | 𝑑𝑡
| ǂ0 driver of B suddenly applies the brakes,
𝑑 |𝑣| 𝑑𝑣 causing his car to deaccelerate at 12 m/s2. It
c) 𝑑𝑡
ǂ 0, while | 𝑑𝑡
|= 0
takes the driver of car A 0.75 s to react. When
45. A particle moves along a straight line so that its he applies his brake, he deaccelerates at 15
velocity depends on time as v = 4t - 𝑡 2 Then for m/s2.Determine the minimum distance d
the first 5s find between the car so as to avoid collision
a) Average velocity
51. Two cars A and B are travelling in the same
b) Average Speed
direction with velocities Va and Vb (Va > Vb).
c) Acceleration
When the car A is at a distance behind car B,
46. A particle moves with an initial velocity Vo and the driver of the car A applies the brakes
retardation αv, where v is velocity at any time t. producing a uniform retardation α, there will be
a) The particle covers a total distance of Vo / α no collision when
b) The particle will come to rest after time 1/ α a) s <
(𝑉𝑎 −𝑉𝑏)2
c) The particle will continue to move for a long 2𝛼
(𝑉𝑎 −𝑉𝑏)2
time b) s = 2𝛼
d) The velocity of particle will become Vo/e (𝑉𝑎 −𝑉𝑏)2
c) s ≥
after time 1/ α 2𝛼
(𝑉𝑎 −𝑉𝑏)2
d) s ≤ 2𝛼

11
UNITS & DIMESNIONS

52. An object constrained to move along the x- axis

58. A body moving with uniform acceleration
travels a distance d1 with constant velocity v1
covers 24 m in the 4th second and 36 m in the
for a time t1. It then instantaneously changes its
6th second. Calculate the acceleration and
velocity to a constant v2 for a time t2 travelling
initial velocity.
a distance d2. Show that
𝑉1 𝑑1 + 𝑉2 𝑑2 𝑉1 𝑡1 + 𝑉2 𝑡2 59. Consider an object moving with an initial
𝑑1 + 𝑑2
≥ 𝑡1 + 𝑡2
velocity of 10 m/s and a = 2m/s². Find distance
53. Consider a particle moving in straight line with travelled from t = 0 to 6 s.
constant acceleration “a” traveling 50 m in 5th 60. Consider a body moving with velocity 9 m/s. It
second and 100 m in 10th second. Find is subjected to acceleration of -2 m/s2.
(a) Initial velocity (u) Calculate the distance travelled by the body in
(b) Acceleration (a) fifth second.
(c) Displacement till 7 s
(d) Velocity after 7 s 61. Acceleration of an object moving in straight line
(e) Displacement between t = 6 s and t = 8 s is a =𝑣 2 and initial velocity of that object is u
m/sec. Find. (i) v(x) i.e. velocity as a function of
54. A point traversed half the distance with a displacement (ii) v(t) i.e. velocity as a function
velocity Vo. The remaining part of the distance of time
was covered with velocity V1 for half the time,
and with velocity V2 for the other half of the 62. Yashwant started moving with constant speed
time. Find the mean velocity of the point 10 m/s to catch the bus. When he was 40 m
averaged over the whole time of motion. away from the bus, it started moving away from
him with acceleration of 2 m/s2. Find whether
55. A car starts moving rectilinearly, first with Yashwant catches the bus or not. If yes, at
acceleration w = 5.0 m/s2 (the initial velocity is what time he catches the bus. If no, then find
equal to zero), then uniformly, and finally, the minimum distance between the bus and
decelerating at the same rate w, comes to a him.
stop. The total time of motion equals t = 25 s.
The average velocity during that time is equal 63. Following information about an object’s motion
to (v) = 72 km per hour. How long does the car is given: a = 𝑡 2
move uniformly? Initial velocity = u
Find: (i) velocity (v) as a function of time.
56. A point moves rectilinearly in one direction. Fig.
(ii) Displacement (x) a function of time.
shows the distance s traversed by the point as
a function of the time t. Using the plot find: 64. The position of an object moving along x-axis
(a) the average velocity of the point during the is given by x = 8.0 + 2.0t2, where x is in meter
time of motion; and t is in second. Calculate:
(b) the maximum velocity; (i) the velocity at t = 0 and t = 2.0 sec.
(c) the time moment to at which the (ii) average velocity between 2.0 sec and 4.0
instantaneous velocity is equal to the mean sec.
velocity averaged over the first to seconds
65. Two tourist A and B who are at a distance of 40
km from their camp must reach it together in
the shortest possible time. They have one
bicycle and they decide to use it in turn. ‘A’
started walking at a speed of 5 km hr–1 and B
moved on the bicycle at a speed of 15 km hr–1.
After moving certain distance B left the bicycle
57. Consider a particle moving in a straight line and walked the remaining distance. A, on
with constant acceleration, has a velocity reaching near the bicycle, picks it up and
Vp = 7 m/s and Vq = 17 m/s, when it crosses the covers the remaining distance riding it. Both
point P and Q respectively. Find the speed of reached the camp together.
the particle at mid-point of PQ. (a) Find the average speed of each tourist.

12
UNITS & DIMESNIONS

(b) How long was the bicycle left unused? c) the velocity is perpendicular to acceleration
66. A particle is moving along positive X direction 72. At t=0; Velocity (u) = 2i + 3j m/s and
and is retarding uniformly. The particle crosses acceleration (a) = 4i + 2j m/s2 respectively. Find
the origin at time t = 0 and crosses the point x the velocity and the displacement of particle at
= 4.0 m at t = 2 s. t = 2 seconds
(a) Find the maximum speed that the particle 73. When the velocity is constant can the average
can possess at x = 0. velocity over any time interval differ from the
(b) Find the maximum value of retardation that instantaneous velocity at any instant? If so,
the particle can have. give an example; if not explain why?
67. A disc arranged in the vertical plane has two
grooves of same length directed along the 74. A particle starts with an initial velocity and
vertical chord AB & CD as shown in the figure. passes successively over the two halves of a
The same particles slide down along AB and given distance with acceleration a1 and a2
CD. The ratio of time tAB / tCD is : respectively. Show that the final velocity is the
A) 1:2 same as if the whole distance is covered with
B) 1: √2 a uniform acceleration: - (a1 +a2) /2
C) 2:1
75. In a car race, car A takes a time t less than car
D) √2 : 1 B at the finish and passes the finishing point
with speed v more than that of car B. Assuming
68. A particle starts moving rectilinearly at time t = that both the cars start from rest and travel with
0 such that its velocity(v) changes with time (t) constant acceleration a1 and a2 respectively.
as per equation – Show that v = √𝑎1 ∗ 𝑎2 t
v = (t 2 – 2t) m/s for 0 < t < 2 s
76. Two particles A and B are connected by a rigid
v = (– t 2 + 6t – 8) m/s for 2 < t < 4 s
rod AB. The rod slides along perpendicular rail
(a) Find the interval of time between t = 0 and t
as shown in the figure. The velocity of A to the
= 4 s when particle is retarding.
left is 10m/s. What is the speed of B when
(b) Find the maximum speed of the particle in
angle Ɵ = 60°?
the interval 0 < t < 4 s.
69. A particle is projected in such a way that it
follows a curved path with constant
acceleration a . For finite interval of motion.
Which of the following option(s) may be correct
u = initial velocity a = acceleration of particle v
= instant velocity for t > 0 77. If x, y and z be the distances moved by a
a) |a X u| ǂ 0 particle with constant acceleration during lth,
b) |a X v| ǂ 0 mth and nth second of its motion respectively,
c) |u X v| ǂ 0 then
d) u. v = 0 a) 𝑥(𝑚 − 𝑛) + 𝑦(𝑛 − 𝑙) + 𝑧(𝑙 − 𝑚) = 0
70. A particle starts moving rectilinearly at b) 𝑥(𝑚 + 𝑛) + 𝑦(𝑛 + 𝑙) + 𝑧(𝑙 + 𝑛) = 0
time t=0 such that its velocity v changes with c) 𝑥(𝑚 − 𝑛) − 𝑦(𝑛 − 𝑙) + 𝑧(𝑙 − 𝑚) = 0
time t according to the equation v = t2-t, d) (𝑚 − 𝑛)𝑦 + (𝑛 − 𝑙)𝑧 + (𝑙 − 𝑚)𝑥 = 0
where t is in seconds and v in s-1. Find the 78. The speed of a train increases at a constant
time interval for which the particle retards. rate 𝛼 from zero to v and then remains constant
for an interval and finally decrease to zero at a
71. Give example where constant rate 𝛽 If b the total distance described,
a) the velocity is in opposite direction to the and t the total time then
𝑙 𝑣 1 1
acceleration a) 𝑡 = 𝑣 + 2 (𝛼 + 𝛽)
b) the velocity of the particle is zero but its 𝑙 𝑣 1 1
acceleration is not zero b) 𝑡 = 𝑣 − 2 (𝛼 + 𝛽)

13
UNITS & DIMESNIONS

𝑙 𝑣 1 1 83. A point moves with uniform acceleration and

c) 𝑡 = 𝑣 − 2 (𝛼 − 𝛽)
𝑙 𝑣 1 1
v1, v2, v3 denote the average velocities in the
d) 𝑡 = 𝑣 + 2 (𝛼 − 𝛽) three successive intervals of time t1, t2, t3.
Which of the following relations is correct?
79. A train stops at two stations s distance apart
and takes time t on the journey from on station a) (v1 - v2) : (v2 - v3) = (t1 - t2) : (t2 + t3)
to the other. Its motion is first of uniform
b) (v1 - v2) : (v2 - v3) = (t1 + t2) : (t2 + t3)
acceleration a and then immediately of uniform
retardation b, then c) (v1 - v2) : (v2 - v3) = (t1 - t2): (t1 - t2) : (t1 - t3)
1 1 𝑡2
a) − =
𝑎 𝑏 𝑠 d) (v1 - v2) : (v2 - v3) = (t1 - t2) : (t2 - t3)
1 1 𝑡2
b) +𝑏 = 𝑠
𝑎 84. A train is moving at a constant speed V when
1 1 𝑡2
c) + = its driver observes another train in front of him
𝑎 𝑏 2𝑠
1 1 𝑡2 on the same track and moving in the same
d) 𝑎
− 𝑏 = 2𝑠 direction with constant speed v if the distance
between trains is x, then what should be the
80. A particle moving along x-axis has acceleration
𝑡 minimum retardation of the train so as to avoid
a at time t given by 𝑎 = 𝑎0 (1 − ) where a0 collision?
𝑇
and T are constants. The particles velocity
when acceleration reduces to zero. a) (V + v)2 / x
1
a) 2 𝑎0 𝑇 2 b) (V - v)2 / x
b) 𝑎0 𝑇 2 c) (V + v)2 / 2x
1
c) 𝑎0 𝑇
2
d) (V - v)2 / 2x
d) 𝑎0 𝑇
85. The declaration experienced by a moving
81. A cone falling with a speed v0 strikes and
motor boat, after its engine is cut off is given by
penetrates the block of a packing material. The
dv/dt= -kv3, where k is constant. If Vo is the
acceleration of the cone after impact is magnitude of the velocity at cut off the
a = g – c x2. Where c is a positive constant and magnitude of the velocity at a time t after the
x is the penetration distance. If maximum cut off is
penetration depth is xm then c equals
a) Vo / 2
2𝑔𝑥𝑚 +𝑣02
a) 2
𝑥𝑚 b) V
2𝑔𝑥𝑚 −𝑣02
b) 2
c) Vo e-kt
𝑥𝑚
𝑉𝑂
6𝑔𝑥𝑚 −3𝑣02 d)
c) 3
2𝑥𝑚
√2 𝑉𝑂2 𝐾 𝑡+1

6𝑔𝑥𝑚 +3𝑣02 86. A train starts from rest and moves with a
d) 3
2𝑥𝑚 constant acceleration for the first 1km. For the
82. a) Can an object have constant velocity and next 3 km, it has a constant velocity and for last
still have a varying speed? 2 km, it moves with constant retardation to
c) Can an object have zero velocity and still come to rest after a total time of motion of 10
be accelerating. min. Find the maximum velocity and the three-
c) Can the average velocity of particle moving time intervals in the three types of motion.
along the x-axis ever be ½ (u + v), if the 87. A point mass moves in a straight line with a
acceleration is not constant. Prove with graph constant acceleration a. At a time t1 after
d) Can the velocity of an object reverse beginning of the motion, the acceleration
direction when its acceleration is constant changes sign remaining the same in
magnitude. Determine the time t from the

14
UNITS & DIMESNIONS

beginning of motion in which the point mass average velocity in this case and s2 is the total
returns to its original position? displacement, Then
a) v2 = 2v1
88. The position vector of a particle varies with time b) 2v1 < v2 <4v1
as r = ro (1 - α t) where ro is a constant vector c) s2 =2s1
and α is a positive constant then the distance d) 2s1 < s2<4s1
covered during the time interval in which
93. For a moving particle which of the following
particle returns to its initial position is:
options are correct
a) ro/ α ⃗⃗⃗⃗⃗⃗
a) |𝑉 𝑎𝑣 | < 𝑉𝑎𝑣
b) ro/ 2 α
⃗⃗⃗⃗⃗⃗
b) |𝑉𝑎𝑣 | > 𝑉𝑎𝑣
𝑟𝑜
c) √𝑟𝑜2 + c) 𝑉 ⃗⃗⃗⃗⃗⃗
𝑎𝑣 = 0, 𝑉𝑎𝑣 ≠ 0
α
⃗⃗⃗⃗⃗⃗
d) 𝑉𝑎𝑣 ≠ 0, 𝑉𝑎𝑣 = 0
2 𝑟𝑜
d) √𝑟𝑜2 + α
94. A particle moves in a straight line with constant
89. A particle moving with uniform acceleration acceleration a. The displacement of particle
along a straight line covers distances a and b from origin in times t1, t2 & t3 are s1, s2 and s3
in successive intervals of p and q seconds. The respectively. If times are in AP with common
acceleration of the particle is difference d & displacements are in GP, then
𝑝𝑞(𝑝+𝑞) prove that a = (√𝑠1 - √𝑠3 )2 /d2
a) 2(𝑏𝑝−𝑎𝑞)
2(𝑎𝑞−𝑏𝑝)
b) 95. To stop a car first you require a certain reaction
𝑝𝑞(𝑝+𝑞)
2(𝑏𝑝−𝑎𝑞) time to begin braking, then the car slows under
c) 𝑝𝑞(𝑝−𝑞) the constant braking de-acceleration. Suppose
2(𝑏𝑝−𝑎𝑞) the total distance moved by the car during
d)
𝑝𝑞(𝑝+𝑞) these two phases is 56.7m when its initial
speed is 80.5 Km/hr and 24.4 m when its initial
90. Velocity of any particle at any time is v =6t i + speed is 48.3 Km/hr. What are your
2j. Find the acceleration and displacement at
t= 2s. Can we apply v = u + at? a) Your Reaction Time
b) Magnitude of de-acceleration
91. A block is dragged on a smooth plane with the 96. A particle is moving along the straight line
help of a rope which moves with velocity v as whose velocity-displacement graph is as
shown in figure. The horizontal velocity of the shown in the figure. What is the magnitude of
block is acceleration when displacement is 3m?

97. A street car moves linearly from station A to

station B with an acceleration varying
92. Starting from rest a particle is first accelerated according to the law f = a – bx where a and b
for time t1 with constant acceleration a1 and are constant & x is the distance from A. The
then stops in time t2 with constant retardation distance between the two stations and the max
a2. Let v1 be the average velocity in this case velocity are
and s1 be the total displacement. In the second a) x = 2a/b, vmax =
𝑎
case it is accelerated for the same time t1 with √𝑏
constant acceleration 2a1 and comes to rest b) x = b/2a, vmax = 𝑏
𝑎
with constant retardation a2 in time t3. If v2 is the

15
UNITS & DIMESNIONS

𝑏 99. Two particles are placed in a gravity free space

c) x = a/2b, vmax =
√𝑎
at (0,0,0) m and (30,0,0) m respectively.
√𝑎 Particle A is projected with a velocity 5i +10j +
d) x = a/b, vmax = 𝑏 5k m/s, while particle B is projected with a
98. A hinged construction consists of three rhombi velocity 10i + 5j + 5k m/s simultaneously. Then
with ratio of sides 3:2:1. Vertex a3 moves in the a) They will collide at (10,20,10) m
horizontal direction at a velocity v. Determine b) They will collide at (10,10,10) m
the velocities of the vertex A1, A2 and B2 at the c) they will never collide
instant when the angles are 90°. d) None of the above
100. Find the relation between acceleration of
blocks a1, a2 and a3

VERTICAL LINEAR MOTION

16
UNITS & DIMESNIONS

EXERCISE - 3

1. A ball is thrown upwards from the top of a tower with constant acceleration 1.2 m/s2. 2s after the
40 m high with a velocity of 10 m/s. Find the start a bolt begins falling from the ceiling of the car.
time when it strikes the ground Find
2. A ball is thrown upwards from the ground with
a) The time after which the bolt hits the floor
an initial speed u m/s. The ball is at the height
of the elevator
of 80 m at two times, the time interval being 6s.
b) The net displacement and distance travelled
Find u
by the bolt, w.r.t to earth
3. A particle is projected vertically upwards with
10. A body is falling freely from a height h above
velocity 40 m/s. Find the distance and
the ground. Find the ratio of distances fallen in
displacement travelled by the particle in
first one second, first two seconds, first three
a) 2s
seconds, also find the ratio of distances fallen
b) 4s
in 1st second,2nd second,3rd second etc.
c) 6s
11. A rocket is fired vertically up from the ground
4. A ball is projected upwards with a speed of 50
with a resultant vertical acceleration of 10m/s2.
m/s. Find the maximum height, time to reach
The fuel is finished in 1 minute and it continues
the maximum height and speed at half the
to move up
maximum height?
a) What is the maximum height reached?
5. A particle is thrown upwards with velocity u =
b) After finishing fuel, calculate the time for
20 m/s. prove that distance travelled in last 2 s
which it continues its upward motion.
is 20 m
12. A block slides down a smooth inclined plane
6. An open lift is moving upwards with velocity
when release from the top while other falls
10m /s. It has an upward acceleration of 2 m/s2.
freely from the same point. Which one of them
A ball is projected upwards with velocity 20 m/s
strike the ground a) earlier b) with greater
relative to ground. Find
speed
a) Time when ball again meets the lift
13. If a body travels half its total path in the last
b) Displacement of lift and ball at that instant
second of its fall from rest, find
c) Distance travelled by the ball up to that
a) Time of its fall
instant
b) Height of its fall
7. An open elevator is ascending with constant
c) Explain the physically unacceptable
speed v =10 m/s. A ball is thrown vertically up
solution of the quadratic time equation
by a boy in the lift when he is at a height of 10
14. A ball is projected vertically up wards with a
m from the ground. The velocity of projection is
velocity of 100 m/s. Find the speed of the ball
v =30 m/s w.r.t to elevator. Find:
at half the maximum height
a) The maximum height attained by the ball
15. A man standing on the edge of a cliff throws a
b) Time taken by the ball to meet the elevator
stone straight up with initial speed u and then
again
throws another stone straight down with the
c) Time taken by the ball to reach the ground
same initial speed and from the same position.
after crossing the elevator
Find the ratio of the speed the stones would
8. From an elevated point A, a stone is projected
have attained when they hit the ground at the
vertically upwards. When the stone reaches a
base of the cliff
distance h below A, its velocity is double of
16. A ball is projected vertically up with an initial
what it was at height h above A. Show that the
speed of 20 m/s and g=10 m/s2. A) How long
greatest height attained by the stone is 5/3 h.
does it take to reach the highest point b) How
9. An elevator car whose floor to ceiling distance
high does it rise above the point of projection
is equal to 2.7 m starts ascending
c) How long will it take for the ball to reach a
point 10m above the point off projection
17. A juggler throws ball into air. He throws one
whenever the previous one is at the highest

17
UNITS & DIMESNIONS

point. How high do the ball rise if he throws n a) 10,20,10

balls each second? b) 5,15,20
18. A body is released from a height and falls freely c) 15,20,15
towards the earth, exactly 1second later d) 5,10,20
another body is released. What is the distance 26. A parachutist drops first freely from an
between the two bodies 2 second after the aeroplane for 10 s and then his parachute open
release of the second body? out. Now he descends with a net retardation of
19. A man in a lift ascending with an upward 2.5 m/s2. If he bails out of the plane at a height
acceleration a throw a ball vertically upwards of 2495 m and g =10m/s2, his velocity on
with a velocity v and catches it after t1 second. reaching the ground will be
After wards when the lift is descending with the a) 5 m/s
same acceleration a acting downwards, the b) 10 m/s
man again throws the ball vertically upwards c) 15 m/s
with the same velocity and catches it after t2 d) 20 m/s
second. Find the velocity and acceleration? 27. A particle is thrown vertically in upward
20. From a lift moving upwards with a uniform direction and passes three equally spaced
acceleration a = 2 m/s2, a man throws a ball windows of equal heights then
vertically upwards with a velocity v =12 m/s a) Average speed of the particles while
relative to the lift. The ball comes back to the passing the windows satisfy the relation
man after a time t. Find the value of t in uav1>uav2> uav3
seconds b) the time taken to cross the window satisfy
21. A balloon rises from the rest on the ground with the relation t1 < t2 < t3
constant acceleration 1 m/s2. A stone is c) The magnitude of the acceleration of the
dropped when the balloon has risen to a height particle while crossing the window satisfy
39.2 m. The time taken by the stone to reach the relation a1 =a2 ǂ a3
the ground (nearly) d) The change in speed of the particle while
22. A body is thrown with a velocity of 100 m/s. It crossing the windows would satisfy the
travels 5m in the last second of its journey. if relation: Δu1 <Δ u2<Δu3
the same body is thrown up with a velocity of 28. A particle slides from rest from the topmost
200 m/s, how much distance will it travel in last point of a vertical circle of radius r along a
second smooth chord making an angle Ɵ with the
23. A particle is dropped from a height h and at the vertical. The time of descent is
same instant another particle is projected a) Least for Ɵ = 0
vertically up from the ground. They meet when b) Maximum for Ɵ = 0
the upper one has descended a height h/3. c) Least for Ɵ = 45°
Find the ratio of their velocities at this instant d) Independent of Ɵ
24. A ball is thrown from the top of the tower in 29. In quick succession a large number of balls are
vertically upward direction. Velocity at a point h thrown up vertically in such a way that the next
m below the point of projection is twice of the ball is thrown up when the previous ball is at
velocity at a point h m above the point of the maximum height. If the maximum height is
projection. Find the maximum height reached 5m, then find the number of balls thrown up per
by the ball above the top of the tower second
a) 2h 30. A lead ball is dropped into a lake from a diving
b) 3h board 5m above the water. It strikes the water
c) 5/3 h with certain velocity and then reaches the
d) 4/3 h bottom with the same constant velocity in 5
25. A juggler keeps on moving the four balls in air seconds after it is dropped. Calculate the
throwing the balls vertically upwards after average velocity of the ball in m/s
regular intervals. When one ball leaves his 31. A stone is dropped from a height
hand (speed = 20 m/s) the position of the other simultaneously another stone is thrown up from
balls (height in metres) will be the ground with such a velocity that it can reach

18
UNITS & DIMESNIONS

a height of 4h. The time when two stones cross ball to return to the ground will be somewhere
each other is between t - a and t +a. Find e and a
ℎ 37. A person standing on the bridge overlooking a
√𝑘𝑔 , where k is _______________
highway inadvertently drops an apple over the
32. At the top of a cliff 100 m high a student throws railing just as the front end of the truck passes
a rock vertically upward with an initial velocity directly below the railing. If the vehicle is
20m/s. How much time later should he drop a moving at 55 Km/hr and is 12m long, how far
second rock from rest so that both the rock above the truck must the railing be if the apple
arrives simultaneously at the bottom of the cliff just misses hitting the rear end of the truck
a) 7 s 38. A rocket is fired vertically and ascends with a
b) 4.5 s constant vertical acceleration of 20 m/s2 for 1.0
c) 4.1 s minute. Its fuel is then all used and it continues
d) 2.5 s as a free fall particle. What is the maximum
33. A steel sphere is released from rest at the altitude reached? What is the total time
surface of a deep tank of viscous oil. A multiple elapsed from take-off time until the rocket
exposure photograph is taken of the sphere as strikes the earth? (Ignore change in g with
it falls. The time interval between exposure is height)
always the same. Which of the following 39. A basketball player about to dunk the ball,
represents this photograph? jumps 76cm vertically. How much time the
34. A man stands on the edge of a cliff. He throws player spends a) in the top 15cm of this jump
a stone upwards with a velocity of 19.6 m/s at and in the bottom 15 cm. Does this explain why
time t = 0. The stone reaches the top of its such players seem to hang in the air at the tops
trajectory after 2s and then falls towards the of their jump?
bottom of the cliff. Air resistance is negligible. 40. A stone is thrown vertically upwards. On its
Which row shows the correct velocity v and way up it passes point A with speed v and point
𝑣
acceleration a of the stone at different times? B, 3 m higher than A with speed . Calculate
2
t/s v/ ms-1
a / ms -2 the speed v and the maximum height reached
A 1.00 9.81 9.81 by the stone above the point B
B 2.00 0 0 41. A woman fell 144 ft from the top of the building
C 3.00 9.81 -9.81 landing on the top of a metal ventilator box
D 5.00 -29.4 -9.81 which she crushed to a depth of 18inch. She
35. An object falls freely with constant acceleration survived without serious injury. What
a from above three light gates. It is found that acceleration did she experience during the
it takes a time t to fall between the first two light collision? Express your answer in terms of g
gates a distance of s1 apart. It then takes an 42. A parachutist after bailing out falls 52 m without
additional time t, to fall between the second friction. When the parachutist opens, she
and the third light gates a distance s2 apart. deaccelerates at 2.10 m/s2 & reaches the
What is the acceleration in terms of s1, s2 and ground with a speed of 2.90 m/s. How long is
t? the parachutist is in the air? At what height did
(𝑠2− 𝑠1 ) the fall begin?
a) 𝑡2
2(𝑠2− 𝑠1 )
43. A bolt is dropped from a bridge under
b) 3𝑡 2
construction, falling 90 m to the valley below
the bridge. (a) In how much time does it pass
(𝑠2− 𝑠1 )
c) through the last 20% of its fall? What is its
2𝑡 2
speed (b) when it begins that last 20% of its fall
d)
2 (𝑠2− 𝑠1 ) and (c) when it reaches the valley beneath the
𝑡2 bridge?
36. A ball is tossed vertically into air with an initial 44. A stone is dropped into a river from a bridge
speed somewhere between (25 - e) m/s and 43.9 m above the water. Another stone is
thrown vertically down 1.00 s after the first is
(25 + e) m/s where e is a small number
dropped. The stones strike the water at the
compared to 25. The total time of flight for the
same time. (a) What is the initial speed of the

19
UNITS & DIMESNIONS

second stone? (b) Plot velocity versus time on after the top of the elevator car passes a bolt
a graph for each stone, taking zero time as the loosely attached to the wall of the elevator
instant the first stone is released. shaft, the bolt falls from rest. (a) At what time
45. To test the quality of a tennis ball, you drop it does the bolt hit the top of the still descending
onto the floor from a height of 4.00 m. It elevator? (b) Estimate the highest floor from
rebounds to a height of 2.00 m. If the ball is in which the bolt can fall if the elevator reaches
contact with the floor for 12.0 ms, (a) what is the ground floor before the bolt hits the top of
the magnitude of its average acceleration the elevator.
during that contact and (b) is the average 52. A catapult launches a test rocket vertically
acceleration up or down? upward from a well, giving the rocket an initial
46. Water drips from the nozzle of a shower onto speed of 80.0 m/s at ground level. The engines
the floor 200 cm below. The drops fall at then fire, and the rocket accelerates upward at
regular (equal) intervals of time, the first drop 4.00 m/s2 until it reaches an altitude of 1 000 m.
striking the floor at the instant the fourth drop At that point, its engines fail and the rocket
begins to fall. When the first drop strikes the goes into free fall, with an acceleration of 29.80
floor, how far below the nozzle are the (a) m/s2. (a) For what time interval is the rocket in
second and (b) third drops? motion above the ground? (b) What is its
47. A steel ball is dropped from a building’s roof maximum altitude? (c) What is its velocity just
and passes a window, taking 0.125 s to fall before it hits the ground? (You will need to
from the top to the bottom of the window, a consider the motion while the engine is
distance of 1.20 m. It then falls to a sidewalk operating and the free-fall motion separately.)
and bounces back past the window, moving 53. A juggler juggles 5 balls with two hands. Each
from bottom to top in 0.125 s. Assume that the ball rises 2m above her hands. Approximately
upward flight is an exact reverse of the fall. The how many times per minute does each hand
time the ball spends below the bottom of the toss a ball?
window is 2.00 s. How tall is the building? 54. What is a reasonable estimate for the
48. Why is the following situation impossible? maximum number of objects a juggler can
Emily challenges David to catch a $1 bill as juggle with two hands if the height to which the
follows. She holds the bill vertically, with the objects are tossed above the hands is h?
centre of the bill between but not touching 55. At the NPL in Delhi a measurement of g was
David’s index finger and thumb. Without made by throwing a glass ball straight up in an
warning, Emily releases the bill. David catches evacuated tube and letting it return. Let ∆𝑡𝐿 be
the bill without moving his hand downward. the time interval between passage across the
David’s reaction time is equal to the average lower level ∆𝑡𝑣 the time interval between the
human reaction time. two passage across the upper level and H is
49. A package is dropped at time t = 0 from a the distance between the two levels. Show that
helicopter that is descending steadily at a 8𝐻
g = (∆𝑡 ) )
speed vi (a) What is the speed of the package 𝐿 2 −(∆𝑡𝑣 2

in terms of vi, g, and t? (b) What vertical

distance d is it from the helicopter in terms of g
and t? (c) What are the answers to parts (a)
and (b) if the helicopter is rising steadily at the
same speed?
50. A ball starts from rest and accelerates at 0.500
m/s2 while moving down an inclined plane 9.00
m long. When it reaches the bottom, the ball
rolls up another plane, where it comes to rest
after moving 15.0 m on that plane. (a) What is 56. A stone, thrown up is caught by the thrower
the speed of the ball at the bottom of the first after 6s. How high did it go and where was it 4
plane? (b) During what time interval does the s after start? g = 9.8 m/s2
ball roll down the first plane? (c) What is the
57. An anti-aircraft shell is fired vertically upwards
acceleration along the second plane? (d) What
is the ball’s speed 8.00 m along the second with a muzzle velocity of 294 m/s. Calculate (a)
plane? the maximum height reached by it, (b) time
51. An elevator moves downward in a tall building taken to reach this height, (c) the velocities at
at a constant speed of 5.00 m/s. Exactly 5.00 s

20
UNITS & DIMESNIONS

the ends of 20th and 40th second. (d) When will carry it to infinity. Find the time it takes in
its height be 2450 m? Given g = 9.8 m/s2. reaching a height h taking the radius of earth
58. A ball is dropped from the roof of a building. An as R and the acceleration due to gravity at the
observer notes that the ball takes 0.1 s to cross surface as g.
over a window 1 m in height. After crossing the 66. An elevator without a ceiling is ascending with
window, the ball takes another 1.00 s to come a constant speed of 6 m/s. A boy on the
to the bottom of the building. What is height of elevator throws a ball directly upward, from a
height of 2.0 m above the elevator floor. At this
the building and how high is the window.
time the elevator floor is 30 m above the
Take g=10ms−2
ground. The initial speed of the ball with
59. A ball thrown up from the ground reaches a respect to the elevator is 9 m/s. (Take g = 10
maximum height of 20 m. Find : (a) Its initial m/s2) (a) What maximum height above the
velocity ; (b) The time taken to reach the ground does the ball reach? (b) How long does
highest point ; (c) Its velocity just before hitting the ball take to return to the elevator floor?
the ground ; (d) Its displacement between 0.5 67. A particle is thrown vertically upwards from the
s and 2.5 s ; (e) The time at which it is 15 m surface of the earth. Let TP be the time taken
above the ground. by the particle to travel from a point P above
60. A balloon starting from the ground has been the earth to its highest point and back to the
ascending vertically at a uniform velocity for 4 point P. Similarly, let TQ be the time taken by
sec and a stone let fall from it reaches the the particle to travel from another point Q
ground in 6 sec. Find the velocity of the balloon above the earth to its highest point and back to
and its height when the stone was let fall. (g = the same point Q. If the distance between the
10 m/s2) points P and Q is H, the expression for
61. A rubber ball is released from a height of 4.90 acceleration due to gravity in terms of TP, TQ
m above the floor. It bounces repeatedly, and H, is: -
always rising to 81/100 of the height through
which it falls. (a) Ignoring the practical fact that 6𝐻
the ball has a finite size (in other words, a) 𝑇𝑝2 + 𝑇𝑄2
treating the ball as point mass that bounces an 8𝐻
b)
infinite number of times), show that its total 𝑇𝑝2 − 𝑇𝑄2
2𝐻
distance of travel is 46.7 m. (b) Determine the c)
𝑇𝑝2 + 𝑇𝑄2
time required for the infinite number of 𝐻
bounces. (c) Determine the average speed d) 𝑇𝑝2 − 𝑇𝑄2
62. A stone is dropped from the top of a tower.
When it crosses a point 5 m below the top, 68. A person sitting on the top of a tall building is
another stone is let fall from a point 25 m below dropping balls at regular intervals of one
the top. Both stones reach the bottom of the second. Find the positions of the 3rd, 4th and 5th
tower simultaneously. Find the height of the ball when the 6th ball is being dropped. Take g
tower = 10 m/s2
63. A particle is dropped from the top of a tower of 69. An elevator car whose floor to ceiling distance
height h and at the same moment, another is equal to 2.7 m starts ascending with constant
particle is projected upward from the bottom. acceleration 1.2 m/s2. Two seconds after the
They meet when the upper one has descended start a bolt begins falling from the ceiling of the
one third of the height of the tower. Find the car. Find: (a) the time after which bolt hits the
ratio of their velocities when they meet and the floor of the elevator.
initial velocity of the lower. (b) the net displacement and distance travelled
64. An elevator whose floor to the ceiling distance by the bolt, with respect to earth. (Take g=9.8
is 2.50 m, starts ascending with a constant m/s2)
acceleration of 1.25 m/s2. One second after the 70. A particle is projected upward from a point A on
start, a bolt begins falling from the ceiling of the the ground. It takes time t1 to reach a point B,
elevator. Calculate: (a) free fall time of the bolt. but it still continues moves up. If it takes further
(b) the displacement and distance covered by t2 time to reach the ground from the point B.
the bolt during the free fall in the reference Then height of point B from the ground is:
frame fixed to the shaft of the elevator. 𝑔 (𝑡 + 𝑡 )2
1 2
65. A particle is projected vertically upwards from a) 2
earth’s surface with a velocity just sufficient to b) 𝑔 𝑡1 𝑡2

21
UNITS & DIMESNIONS

𝑔 (𝑡1 + 𝑡2 )2 78. A body dropped from the top of a tower covers

c) 8
1
7/16 of the total height in the last second of its
d) 𝑔 𝑡1 𝑡2 fall. The time of fall is
2

71. Two different balls of masses m1 & m2 are 79. A particle is thrown up inside a stationary lift of
allowed to slide down from rest and from same sufficient height. The time of flight is T. Now it
height h along two smooth inclined planes is thrown again with same initial speed V0 with
having different angles of inclination α and β. respect to lift. At the time of second throw, lift is
Then the moving up with speed V0 and uniform
a) The final speed acquired by them will be acceleration g upward (the acceleration due to
the same gravity). The new time of flight is
b) The final speed acquired by them will be 80. A coin is released inside a lift at a height of 2 m
different from the floor of the lift. The height of the lift is
c) The time taken by them to reach the bottom 10 m. The lift is moving with an acceleration of
will be the same 9 m/s2 downwards. The time after which the
d) Time taken by them to reach the bottom will coin will strike with the lift is: (g = 10 m/s2)
be in the ratio (sin β / sin α)
81. A ball is thrown vertically upwards with an initial
72. A ball starts falling with zero initial velocity on a velocity of 5 m/sec from point P as shown. Q is
smooth inclined plane forming an angle a with a point 10 m vertically below the point P. Then
the horizontal. Having fallen the distance h, the the speed of the ball at point Q will b: (take g =
ball rebounds elastically off the inclined plane. 10 m/s2 and neglect air resistance)
At what distance from the impact point will the
-ball rebound for the second time?

73. A body is projected from the bottom of a

smooth inclined plane with a velocity of 20 m/s.
If it is just sufficient to carry it to the top in 4s,
find the inclination and height of the plane.
82. A stone is thrown straight upward with a speed
74. A ball is dropped from an elevator at an altitude of 20 m/s. It is caught on its way down at a point
of 200 m. How much time will the ball take to 5.0 m above where it was thrown. (a) How fast
reach the ground if the elevator is was it going when it was caught? (b) How long
a) Stationary did the trip take?
b) ascending with velocity 10 m/s?
c) descending with velocity 10 m/s? 83. A rope is lying on a table with one of its end at
point O on the table. This end of the rope is
75. A particle is projected vertically upwards. pulled to the right with a constant acceleration
Prove that it will be at 3/4 of its greatest height starting from rest. It was observed that last 2 m
at times which are in the ratio 1:3. length of the rope took 5 s in crossing the point
76. Two nearly identical balls are released O and the last 1m took 2 s in crossing the point
simultaneously from the top of a tower. One of O.
the balls fall with a constant acceleration of g1 (a) Find the time required by the complete
= 9.80 ms–2 while the other falls with a constant rope to travel past point O.
acceleration that is 0.1% greater than g1. [This (b) Find length of the rope.
difference may be attributed to variety of
reasons. You may point out few of them]. What
is the displacement of the first ball by the time
the second one has fallen 1.0 mm farther than 84. A ball is projected vertically up from the ground
the first ball? surface with an initial velocity of u = 20 m/s. O
77. A pebble is thrown horizontally from the top of is a fixed point on the line of motion of the ball
a 20 m high tower with an initial velocity of 10 at a height of H = 15 m from the ground. Plot a
m/s. The air drag is negligible. The speed of the graph showing variation of distance (s) of the
pebble when it is at the same distance from top ball from the fixed-point O, with time (t). [Take
as well as base g = 10 m/s2]. Plot the graph for the entire time
of the tower (g = 10 m/s2) of flight of the ball.

22
UNITS & DIMESNIONS

ball at regular intervals of Δt = 1.0 s after the

throw. (a) How many drops will detach from
the ball before it hits the ground. (b) How far
away the drops strike the ground from the point
where the ball hits the ground?
90. Two stones of mass m and M (M > m) are
dropped Δt time apart from the top of a tower.
Take time t = 0 at the instant the second stone
85. A prototype of a rocket is fired from the ground. is released. Let Δv and Δs be the difference in
The rocket rises vertically up with a uniform their speed and their mutual separation
5
acceleration of m/s2.8 second after the start a respectively. Plot the variation of Δv and Δs
4 with time for the interval both the stones are in
small nut gets detached from the rocket.
flight. [g = 10 m/s2]
Assume that the rocket keeps rising with the
constant acceleration. 91. Two bodies start to fall from the same height at
(a) What is the height of the rocket at the an interval of y seconds. After what time t since
instant the nut lands on the ground the first body started, their separation will be d.
(b) Plot the velocity – time graph for the motion
of the nut after it separates from the rocket till 92. Let 𝑟⃗(t) be the position vector of a particle of
𝑑
it hits the ground. Plot the same velocity– time fixed magnitude. Show that 𝑑𝑡 𝑟⃗ (t) is
graph in the reference frame of the rocket. perpendicular to 𝑟⃗(t).
Take vertically upward direction as positive and
g = 10 m/s2 93. Show that the acceleration of a particle which
86. An elevator starts moving upward with constant travels along the space curve with velocity is
acceleration. The position time graph for the given by
𝑑𝑣 𝑣2
floor of the elevator is as shown in the figure. a= 𝑑𝑡
𝑇̂ + 𝜌
̂
𝑁
The ceiling to floor distance of the elevator is
where 𝑇̂ is the unit Tangent vector to the space
1.5 m. At t = 2.0 s, a bolt breaks loose and
̂ is its unit principal Normal and 𝜌 is the
curve, 𝑁
drops from the ceiling. (a) At what time t0 does
the bolt hit the floor? (b) Draw the position time radius of curvature?
graph for the bolt starting from time t = 0. [take 94. A particle of mass M is thrown vertically upward
g = 10 m/s2] with a velocity u from the earth’s surface. If the
frictional force is mkv2, where v is the
instantaneous velocity and k is the force
constant show that the time taken by the
particle to reach the maximum height is given
by
1 𝐾
𝑇= tan−1 𝑢√
√𝑔𝐾 𝑔
95. A smooth straight wire of length l has its upper
87. At t = 0 a projectile is projected vertically up end attached to the top O of a vertical pole
with a speed u from the surface of a peculiar making an angle 30°. A bead is allowed to slip
planet. The acceleration due to gravity on the along the wire from the top to the bottom. Find
planet changes linearly with time as per the velocity of the bead when it reaches the
equation g = at where a is a constant. free end
88.
(a) Find the time required by the projectile to
attain maximum height.
(b) Find maximum height attained.
(c) Find the total time of flight.
89. A wet ball is projected horizontally at a speed
of u = 10 m/s from the top of a tower h = 96. A tennis ball fall freely from a height H on an
31.25 m high. Water drops detach from the inclined plane making an angle of 45° with the

23
UNITS & DIMESNIONS

horizon. After bouncing the ball falls on the

Where k is resistive force constant
plane again. Prove that the distance between
98.
two points of striking is 4𝐻√2.
97. A particle is projected vertically upwards with a
velocity Vo. Assuming air resistance to be
proportional to the velocity of the particle, show
that the maximum height attained by it is given
by
𝑉 𝑔 𝑔
Xm = 𝐾𝑜 + 𝑘 2 ln ( 𝑔+𝑘 𝑉 )
𝑜

GRAPHS

24
UNITS & DIMESNIONS

RELATIVE VELOCITY

Case -1:

Case -2:

Case -3:

25
UNITS & DIMESNIONS

Case -4: River- Swimmer Problem

Case-5: River -Swimmer Problem

Case-6: Wind – Plane Problem

26
UNITS & DIMESNIONS

Case-7: Wind – Plane Problem

Case-8: Up-Stream and Down-Stream in River

Case-9: Linear Escalator Problem

27
UNITS & DIMESNIONS

Case-10: Rain Problem

28
UNITS & DIMESNIONS

EXERCISE - 4

Based on X-T GRAPHS

1. Plot the x-t graph for constant velocity and constant acceleration
2.

29
UNITS & DIMESNIONS

30
UNITS & DIMESNIONS

31
UNITS & DIMESNIONS

LEVEL -1 2. An athlete completes one round of a circular

1. A man goes 10 m towards North, then 20 m track of radius R in 40 second. What will be his
towards east then displacement displacement at the end of 2 min 20 second?
(a) 22.5 m (a) Zero
(b) 25 m (b) 2R
(c) 25.5 m (c) 2 pie R
(d) 30 m (d) 7 pie R
3. A person travels along a straight road for half
the distance with velocity V1 and the remaining

32
UNITS & DIMESNIONS

half distance with velocity V2. The average (a) 20 m /s2

velocity is given by (b) – 20 m /s2
(a) 𝑉1 𝑉2 (c) - 40 m/ s2
𝑉22
(b) (d) +2 m / s2
𝑉12
𝑉1 + 𝑉2 9. Which of the following four statements is
(c) false:-
2
2 𝑉1 𝑉2
(d) (a) A body can have zero velocity and still be
𝑉1 + 𝑉2
accelerated
4. A 150 m long train is moving with a uniform
velocity of 45 km/h. the time taken by the train (b) A body can have a constant velocity and
to cross a bridge of length 850 meters is still have it varying speed
(a) 56 second (c) A body can have a constant speed and
(b) 68 sec still have a varying velocity
(c) 80 sec (d) The direction of the velocity of a boy can
(d) 92 sec change when its acceleration is constant
5. Which of the following options is correct for the 10. A car moving with a velocity of 10 m/s can be
object having a straight- line motion stopped by the application of a constant force
represented by the following graph: F in a distance of 20 m. If the velocity of the
car is 30 m/s, it can be stopped by this force
in
(a) 20/3 m
(b) 20 m
(c) 60 m
(d) 180 m
11. The displacement of a particle is proportional
(a) The object moves with constantly to the cube of time elapsed. How does the
increasing velocity from O to A and then it acceleration of the particle depend on time
obtained?
moves with constant velocity.
(a) a ∝ t2
(b) Velocity of the object increases uniformly.
(b) a∝t4
(c) Average Velocity is zero
(c) a∝t3
(d) The graph shown in impossible.
(d) a∝t
6. A particle experiences a constant acceleration
for 20 sec after starting from rest. If it travels a 12. A stone falls from a balloon that is descending
distance S1 in the first 10 sec and a distance at a uniform rate of 12 m/s. the displacement of
S25 in the next 10 sec, then the stone from the point of release after 10 sec
(a) S1 = S is
(a) 490 m
(b) S1 =S2 / 3
(b) 510 m
(c) S1 =S2 / 2
(c) 610 m
(d) S1 = S2 / 4
(d) 725 m
7. A body under the action of several forces will
have zero acceleration. 13. A body A is projected upwards with a velocity
(a) When the body is very light of 98 m/s. The second body B is projected
(b) When the body is very heavy upwards with the same initial velocity but after
4 sec both the bodies will meet after
(c) When the body is a point body
(a) 6 sec
(d) When the vector sum of all the forces
(b) 8 sec
acting on it is zero
(c) 10 sec
8. A motor car moving with a uniform speed of 20
(d) 12 sec
m/sec comes to stop on the application of
14. A ball P is dropped vertically and another ball
brakes after travelling a distance of 10 m its
Q is thrown horizontally with the same
acceleration is

33
UNITS & DIMESNIONS

velocities from the same height and at the (a) 60 m

same time. If air resistance is neglected then (b) 55 m
(a) Ball P reaches the ground first (c) 25 m
(b) Ball Q reaches the ground first (d) 30 m
(c) Both reach the ground at the same time
(d) The respective masses of the two balls will 19. The graph between the displacement x and time
decide the time for a particle moving in a straight line is shown
15. A stone dropped from the top of the tower figure. During the interval OA, AB, BC and CD, the
touches the ground in 4 sec. The height of the acceleration of the particle is
tower is about
(a) 80 m
(b) 40 m
(c) 20 m
(d) 160 m
16. Time taken by an object falling from rest to
cover the height of h1 and h2 is respectively t1
and t2 then the ratio of their time is
(a) h1:h2
(b) √ℎ1 : √ℎ2
(c) h1 : 2h2
(d) 2h1 : h2 OA AB BC CD
17. A particle starts from rest its acceleration (a) a) + 0 + +
versus time (t) s as shown in the figure the b) - 0 + 0
c) + 0 - +
maximum speed of the particle will be
d) - 0 - 0
(a) 10 m/s
(b) 55 m/s
(c) 550 m/s
(d) 600 m/s 20. A lift is going up. The variation in the speed of
the lift is as given in the graph. What is the
height to which the lift takes the passengers?

18. The variation of velocity of a particle with time

moving along a straight line is illustrated in the
following figure. The distance travelled by the (a) 3.6 m
particle in four seconds is (b) 28.8 m
(c) 36.0 m
(d) can not be calculated from the above
graph
21. The displacement time graph of moving
particle is shown below

34
UNITS & DIMESNIONS

The instantaneous velocity of the particle is

negative at the point
(a) D
(b) F
(c) C
(d) F LEVEL - 2
22. A train moves from one station to another in 2 1. A particle covers half of the circle to radius r.
hours’ time its speed time graph during this Then the displacement and distance of the
motion is shown in the figure. The maximum particle are respectively.
acceleration during the journey is (a) 2πr, 0
(b) 2r, π r
(c) πr/ 2, 2r
(d) πr, r
2. A hall has the dimension 10 m× 10 m×10 m. A
fly starting at one corner ends up at a
diagonally opposite corner. The magnitude of
its displacement is nearly
(a) 5 √3 m
(a) 140 km h2
(b) 10 √3 m
(b) 160 km h2
(c) 20 √3 m
(c) 100 km h2
(d) 30 √3 m
(d) 120 km h2
3. A car travels from A to B at a speed of 20 km
23. The acceleration time graph of a body is
h-1 and returns at a speed of 30 km h-1. The
shown below
average speed of the car for the whole
journey is:-
(a) 5 km h-1
(b) 24 km h-1
(c) 25 km h-1
(d) 50 km h-1
4. A car travels a distance of 2000 m. if the first
half distance is covered in 40 km/hour and the
second half at velocity v and if the average
velocity is 48 km/h. then the value of v is:
The most probable velocity time graph of the (a) 56 km/h
body is (b) 60 km/h
(c) 50 km/h
(d) 48 km/h
5. At any instant t, the co-ordinates of particle
are x =at2, y = bt2 & z = 0, then its velocity at
the instant t will be

35
UNITS & DIMESNIONS

a) 𝑡 √𝑎2 + 𝑏 2 (d) 75 m
b) 2𝑡 √𝑎2 + 𝑏 2 11. A body starts from rest, the ratio of a distance
travelled y the body during 3rd and 4th seconds
c) √𝑎2 + 𝑏 2
is
d) 2 t2 √𝑎2 + 𝑏 2 (a) 7/5
6. A car runs at constant speed on a circular (b) 7/9
track of radius 100 m taking 62.8 s on each (c) 7/3
lap. What is the average speed and average (d) 3/7
velocity on each complete lap?
12. Two trains each of length 50 m are
a) Velocity10 m/s, speed 10 m/s
approaching each other on parallel rails. Their
b) Velocity zero, speed 10m/s velocities are 10 m/s and 15 m/s. They will
c) Velocity zero, speed zero cross each other in
d) Velocity 10 m/s, speed zero (a) 2 sec
7. The displacement of a body is given by (b) 4 sec
2s=gt2 where g, I, s & a are constant. The (c) 10 sec
velocity of the body at any time t is: - (d) 6 sec
(a) gt
13. A particle after starting from rest, experience,
(b) gt/2 constant acceleration for 20 s, if it covers a
(c) gt2/2 distance of S1, in first 10 seconds and
(d) gt3/3 distance S2 in next 10 sec the.
8. The displacement time graph of a moving (a) S2=S1/2
particle is shown below. The instantaneous (b) S2=S1
velocity of the particle is negative at the point. (c) S2=2S1
(d) S2=3S1
14. A body sliding on a smooth inclined plane
requires 4 sec to reach the bottom after
starting from rest at the top. How much time
does it take to cover one fourth the distance
starting from the top
(a) 1 sec
(b) 2 sec
(a) C
(c) 0 sec
(b) D
(d) 1.6 sec
(c) E
(d) F
9. A body starts from rest and is uniformly
accelerated for 30 s. the distance travelled in 15. A body is dropped from a height h under
the first 10 s is x1, next 10 s is x2 and the last acceleration due to gravity g. if t1 and t2 are
10 s is x3. Then x1 : x2 : x3 the same as time intervals for its fall for first half and the
(a) 1:2:4 second half distance, the relation between
(b) 1:2:5 then is
(c) 1:3:5 (a) t1 = t2
(d) 1:3:9 (b) t1 = 2t2
10. The initial velocity of a particle is 10 m/s and (c) t1 = 2.414 t2
its retardation is 2 m/sec square. The distance (d) t1 = 4t2
covered in the fifth second of the motion will 16. Two bodies of different masses ma and mb
be are dropped from two heights a and b.The
(a) 1 m ratio of times taken by the two to drop through
(b) 19 m these distance is:
(c) 50 m a) a : b

36
UNITS & DIMESNIONS

b) ma : mb = b:a time axis. If the velocity of A is VA and that of

c) √𝑎 : √𝑏 B is VB then the value of VA / VB is
1
d) a2 : b2 (a) 2
17. A body is thrown upward and reaches its (b)
1
maximum height. At that position. √3
(a) Its velocity is zero and its acceleration is (c) √3
1
also zero (d)
3
(b) Its velocity is zero but its acceleration is 22. The v- t graph of a linear motion is shown in
maximum adjoining figure. The distance from origin after
(c) Its acceleration is minimum 8 seconds is
(d) Its velocity is zero and its acceleration is
the acceleration due to gravity.
18. A ball is thrown upwards. It returns to ground
describing a parabolic path. Which of the
following remains constant?
(a) Speed of the ball
(b) Kinetic energy of the ball
(c) Vertical component of velocity
(d) Horizontal component of velocity
19. A particle is moving so that its displacement (a) 18 m
is given as s= t3 - 6t2 + 3t+ 4 meter. Its velocity (b) 16 m
at the instant when its acceleration is zero will (c) 8 m
be: - (d) 6 m
(a) 3 m/s 23. The adjoining curve represent the velocity
(b) -12 m/s time graph of a particle, its acceleration value
(c) 42 m/s along OA, AB and BC m/second square are
(d) -9 m/s respectively.
20. The variation of velocity of a particle moving
along straight line is shown in the figure. The
distance travelled by the particle in 4 s is
(a) 25 m
(b) 30 m
(c) 55 m
(d) 60 m

(a) 1, 0, -0.5
(b) 1, 0, 0.5
(c) 1, 1, 0.5
(d) 1, 0.5, 0
24. In the following velocity time graph of a body
the distance and displacement travelled by
the body in 5 second in meters will be:-

21. The displacement time graphs of two particle

A and B are straight lines making angles of
respectively 30 degree and 60 degree with the

37
UNITS & DIMESNIONS

2. The coordinates of a moving particle at a time

t are given by x=5 sin 10 t, y=5 cos 10 t. The
speed of the particle is:-
(a) 25
(b) 50
(c) 10
(d) none
3. A particle starts moving rectilinearly time t=0
(a) 75,115 such that its velocity ‘v’ changes with time ‘t’
(b) 105,75 according to the equation v=t square-t, where
(c) 45,75 t is in seconds and v is in m/s. The time
(d) 95,55 interval for which the particle retards is
25. If position time graph of a particle is sine (a) t<1/2
curve as shown, what will be its velocity time (b) ½ <t<1
graph. (c) t>1
(d) t<1/2 and t>1
4. A stone is dropped into a well in which the
level of water is h below the top of the well. If
v is the velocity of sound, the time after which
splash is heard is given by
2ℎ
a) T =
𝑣
2ℎ ℎ
b) T = √ 𝑔 + 𝑣

2ℎ ℎ
c) T = √ 𝑔 + 2𝑣

ℎ 2ℎ
d) T = √ +
2𝑔 𝑣
5. A ball is thrown vertically down with velocity of
5 m/s. With what velocity should another ball
be thrown down after 2 seconds so that it can
hit the 1st ball in 2 seconds
(a) 40 m/s
(b) 55 m/s
(c) 15 m/s
(d) 25 m/s
6. A particle is projected vertically upward from a
point A on the ground. It takes time t1 to reach
a point B but it still continues to move up. If it
LEVEL 3 takes to further time t2 to reach the ground
1. A body covers first 1/3 part of its journey with a from point B, then height of point B from the
velocity of 2 m/s, next 1/63 part with a velocity ground is:
1
of 3 m/s and rest of the journey with a velocity a) 2
𝑔 (𝑡1 + 𝑡2 )2
6 m/s. The average velocity of the body will be:- 1
b) 𝑔 𝑡1 𝑡2
(a) 3 m/s 2
(b) 11/3 m/s c) 𝑔 𝑡1 𝑡2
1
(c) 8/3 m/s d) 8
𝑔 (𝑡1 + 𝑡2 )2
(d) 4/3 m/s 7. Balls are thrown vertically upwards in such a
way that the next ball is thrown when the
previous one is at the maximum height. If the

38
UNITS & DIMESNIONS

maximum height is 5 m, the number of balls (d) The information is sufficient to decide the
thrown per minute will be:- relation
(a) 40 (b) 50 (c) 60 (d) 120 13. Acceleration versus velocity graph of a particle
8. A disc arranged in a vertical plane has two moving in a straight-line starting form rest is as
groves of same length directed along the shown in figure. The corresponding velocity
vertical chord AB and CD as shown in the time graph would be:-
figure. The same particle slide down along AB
and CB. The ratio of the time tAB / tCD is.

a) 1 : 2
b) 1 : √2
c) 2 : 1
d) √2 ∶ 1
9. A body moves with velocity v=lnx m/s where x
is its position. The net force acting on body is 14. A man moves in x-y plane along the path
zero at. shown. At what point is his average velocity
(a) 0 m vector in the same direction as his
(b) x=e2 m instantaneous velocity vector. The man starts
(c) x= e m from point P.
(d) x=1m
10. A body of mass 1 kg is acted upon by a force
F = 2 sin 3πt i + 3 cos 3 πtj. Find its position
at t=1 s, if at t=0s it is at origin and at rest
3 3
a) 3𝜋2
, 9𝜋2
2 2
b) 3𝜋2
, 9𝜋2
c)
4 5
, 9𝜋2 (a) A
3𝜋2
(b) B
d) None of the above
(c) C
11. A force F=Be-C1 acts on a particle whose mass
is m and whose velocity is 0 at t=0. Its terminal (d) D
velocity (velocity after a long time) is:-
Q15. The acceleration of a particle which moves
(a) C/mB
along the positive x-axis varies with its position as
(b) B/mC
shown. If the velocity of the particle is 0.8 m/s at
(c) BC/m
x=0, the velocity of the particle at x=1.4 m/x
(d) –B/mC
12. A particle has a velocity u towards east at t=0.
Its acceleration is towards west and is
constant, Let XA and Xn be the magnitude of
displacement in the first 10 seconds and next
10 seconds
(a) XA < Xn
(b) XA = Xn
(c) XA > Xn

39
UNITS & DIMESNIONS

(a) 1.6 3. A particle is moving along x-axis. Initially is it

(b) 1.2 located 5 m left of origin and it is moving away
(c) 1.4 from the origin and slowing down. In the
(d) none coordinate system, the signs of the initial
velocity and acceleration are
Q16. The velocity time graph of a body falling from
rest under gravity and rebounding from a solid
surface is represented by which of the following
graphs?

4. A speeder in a automobile passes a

stationary policeman who is hiding behind a
ball board with a motorcycle after a 2.0 sec
delay (reaction time) the policeman
accelerates to his maximum speed of 150
km/hr in 12 sec and catches the speeder 1.5
km beyond the billboard. Find the speed of
speeder in km/hr.
5. At a distance I-400 m from the traffic light
brakes are applied to a locomotive moving at
a velocity v=54 km/hr. Determine the position
of the locomotive relative to the traffic light 1
min after the application of the breaks if ts
acceleration is -0.3 m/ second square.
Q17. Shown in the figure are the velocity time 6. A stone is dropped from a height h.
graphs of the two particles P1 and P2. Which of the Simultaneously another stone is thrown up
following statements about their relative motion is from the ground with such a velocity that it
true? Their relative velocity can reach a height of 4 h. Find time when two
stones cross each other.
7. Velocity of car v is given by v=at-bt2, where a
and b are positive constants & t is time
elapsed. Find value of time for which velocity
is maximum & also corresponding value of
velocity.
8. The force acting on a body moving in a
straight line is given by F= (3t2-4t+1) Newton
(a) Is zero where t is in sec. if mass of the body is 1 kg
(b) Is non zero but constant and initially is was at rest at origin. Find
(c) Continuously decrease displacement between time t=0 and t=2
(d) Continuously increase second (use F=ma)
9. The velocity time graph of a body moving in
a straight line is shown. Find its

LEVEL- 4
1. A particle goes from A to B with a speed of
40 km/h and B to C with a speed of 60 km/h.
If AB=6BC the average speed in km/h
between A and C is _________.
2. Find the change in velocity of the tip of the
minute hand (radius= 10 cm) of a clock in 45
minutes. (a) Instantaneous velocity at=1.5 sec

40
UNITS & DIMESNIONS

(b) Average acceleration from t=1.5 sec to 1. A truck driver travels three fourths of the
t=2.5 sec distance of his run at constant velocity (v) and
(c) Draw its acceleration time graph from t=0 then travels the remaining distance at velocity
to t=2.5 sec. of (v/2). What was the truckers average speed
10. A train starts from rest and moves with a for the trip?
constant acceleration of 2.0 m/s square for 2. Men are running along a road at 15 km/hr
half a minute. The breaks are then applied behind one another at equal intervals of 20 m.
and the train comes to rest in one minute. Cyclists are riding in the same direction at 25
Find km/h at equal intervals of 30 m. At what speed
(a) The total distance moved by the train an observer travel along the road in opposite
(b) The maximum speed attained by the direction so that whenever he meets a runner,
train and he also meets a cyclist? (Neglect the size of
cycle)
(c) The position (s) of the train at half the
3. An object moving with uniform acceleration has
maximum speed
a velocity of 12.0 cm/s in the positive x direction
11. From the velocity time plot shown in figure. when its x coordinate is 3.00 cm. if x-coordinate
Find the distance travelled by the particle
2.00 later is -5.00 cm, what is its acceleration?
during the first 40 seconds. Also find the
4. A ball is thrown vertically upwards from the
average velocity during this period.
ground. It crosses a point at the height of 25 m
twice an interval of 4 sec. With what velocity
the ball was thrown?
5. From the top of a tower, a ball is thrown
vertically upwards. When the ball reaches h
below the tower, its speed is double of what it
was at height h above the tower. Find the
greatest height attained by the ball from the
12. The velocity time graph of the particle tower.
moving along a straight line is shown. The 6. A man in a balloon rising vertically with an
rate of acceleration and deceleration is acceleration of 5 m/s square release a ball 25
constant and it is equal to 5 ms-2. If the seconds after the balloon is let go from the
average velocity during the motion is 20 ms- ground. Find the greatest height above the
1
, then find the value of t. ground reached by the ball.
7. Force acting on a body of mass 1 kg is related
to its position x as F= x3 - 3x N. it is at rest x=1.
What is its speed at x=3? (Use F=ma)
8. Figure gives the speed time graph of the
motion of a car. What is the ratio of the distance
travelled by the car during the last two seconds
13. The figure shows the v-t graph of a particle to the total distance travelled in seven
moving in straight line. Find the time when seconds?
particle returns to the starting point.

9. The velocity time graph of the particle moving

along a straight line is shown. The rate of
LEVEL - 5 acceleration and deceleration is constant and
it is equal to 5 ms2, if the average velocity

41
UNITS & DIMESNIONS

during the motion is 20 ms1, then find the value at time t=0 is x=0. Find displacement ( in m) of
of t? particle in 2 sec.
Q4. A man walking from town A to another town B
at the rate of 74 km/hr starts one hour before a
coach (also travelling from A to B). The coach is
travelling at the rate of 12 km/h and on the way
he is picked up by the coach. On arriving B, he
10. Acceleration of particle moving in straight line finds that his coach journey lasted 2 hours. Find
can be written as from the given graph find the distant in km between A and B.
acceleration at x=20m. Q5. A train is travelling at v m/s along a level
straight track. Very near and parallel to the track
is a wall. On the wall a naughty boy has drawn a
straight line that slopes upward at a 37 degree
angle with the horizontal. A passenger in the train
is observing the line out of window (0.90 m high ,
1.8 m wide as shown in figure). The line first
appears at window corner A and finally
disappears at window corner B. if it takes 0.4 sec
JEE ADVANCED INTEGER TYPE
between appearance at A and disappearance of
Q1. A high speed Jet starts from rest at s=0 and the line at B, What is the value of v in cm/s
is subjected to the acceleration shown in the
figure. Determine the velocity of the after it has
travelled 50 m.

Q6. Acceleration of particle moving rectilinearly is

a=4-2x (where x is position in metre and a in m/s
square). It is at instantaneous rest at x=0. At what
Q2. A boy throws a ball with speed u in a well of position x in meter will the particle again come to
depth 14 m as shown. On bounce with bottom of instantaneous rest?
the well the speed of the ball gets halved. What
Q7. A cricket ball moving along ground in straight
should be the minimum value of u in m/s such
line goes past Yuvraj at a constant speed of 5
that the ball may be able to reach his hand again?
m/s. Yuvraj starts from rest and accelerates at a
It is given that his hands are at 1 m height from
constant rate of 2 m/s square at the instant ball
top of the well while throwing and catching
crosses him. What is speed of Yuvraj in m/s as he
catches the ball.
Q8. The figure shows the graph of velocity time
for a particle moving in a straight line. If the
average speed for 6 sec is ‘b’ and the average
acceleration from zero second to 4 second is ‘c’
find magnitude of be in m square/ second square

Q3. Velocity of a particle moving in a straight line

varies with its displacement as v= root 4+4x m/s
where x is displacement. Displacement of particle

42
UNITS & DIMESNIONS

every 5 s such that it always leaves her hand

with speed 2 ms-1 with respect to the ground.
Consider two cases:
(a) P runs with speed 1 ms-1 towards Q while
Q remains stationary.
(b) Q runs with speed 1 ms-1 towards P while
P remains stationary

Q9.A rock is shot vertically upward from the edge Note that irrespective of speed of P, ball always
of the top of a tall building the rock reaches its leaves3` P’s hand with speed 2 ms-1 with
maximum height above the top of building 1.75 respect to the ground. Ignore gravity. Balls will
after being shot , then after barely missing the be received by Q
edge of the building as it falls downward, the rock a) One every 2.5 in case (a) and one
strikes the ground 6.0 s after it is launched . In SI every 3.3 in case (b)
units, how tall is the building?
b) One every 2 s in case (a) and one
Q10. A stone is thrown upwards with an initial every 4 s in case (b)
speed of 10 m/s while standing on the edge of a c) One every 3.3 in vase (a) and one
cliff. Find the distance travelled in m by the stone every 2.5 in case (b)
till 5 second. d) One every 2.5 in case (a) and one
KVPY (PREVIOUS YEARS) every 2.5 in case (b)
4. The accompanying graph of position x versus
1. A girl standing at point P on a beacgh wishes time t represents the motion of a particle. If p
to reach a point Q in the sea as quickly as and q are both positive constants, the
possible. She can run at 6 km h-1 on the expression that best describes the acceleration
beach and swim at 4 km h-1 in the sea. She alpha of the particle is
she should take the path
(a) PAC
(b) PBQ
(c) PCQ
(d) PDQ

(a) A= -pp –at

(b) a= - p+qt
(c) a=p+qt
(d) a=p-qt
2. A juggler tosses a ball up in the air with initial
5. In the following displacement (x) vs time (t)
speed u. At the instant it reaches its maximum
height H, he tosses up a second ball with the graph at which among the P, Q and R is the
objects speed increasing?
same initial speed. The two balls will collide at
a height.
(a) H/4
(b) H/2
(c) 3H/4
3
(d) √4 H
3. Two skaters P and Q are skating towards (a) R only
each other. Skater P throws a ball towards W (b) P only

43
UNITS & DIMESNIONS

(c) Q and R only

(d) P,Q, R

6. An object at rest at the origin begins to move

in the +x direction with a uniform acceleration
of 1 m/s square for 4 s and then it continue
moving with a uniform velocity of 4 m/s in the
same direction. The x-t graph for objects
motion will be:-

7. A boy is standing on top of a tower of height

85 m and throws a ball in the vertically upward
direction with a certain speed. If 5.25 seconds
later he hears the ball hitting the ground, then
the speed with which the boy threw the ball is JEE MAIN PREVIOUS YEARS
(take g=10 m/s square, speed of sound in air
340 m/s) 1. An object moving with a speed of 6.25 m/s, is
(a) 6 m/s decelerated at a rate given by:- dv/dt= -2.5 √𝑣
(b) 8 m/s where v is the instantaneous speed. The time
(c) 10 m/s taken by the object, to come to rest would be
(d) 12 m/s (a) 1 s
8. If a ball is thrown at a velocity of 45 m/s in (b) 2 s
vertical upward direction , then what would be (c) 4 s
the velocity profile as function of height? (d) 8 s
Assume g= 10 m/s2
2. From a tower of height H, a particle is thrown
vertically upwards with speed u. The time
taken by the particle, to hit the ground, is a n
times that taken by it to reach the highest
point of its path.
(a) 2 g H=n2u2
(b) gH= (n-2)2 u2
(c) 2g H= nu2(n-2)
(d) gH=(n-2) u2
3. A body is thrown vertically upwards. Which
one of the following graphs correctly represent
the velocity vs time?

44
UNITS & DIMESNIONS

the bullet after emerging from the other side of

the wall is close to:-
(a) 0.4 m s-1
(b) 0.1 ms-1
(c) 0.3 m s-1
(d) 0.7 ms-1
7. A particle starts from origin O from rest and
moves with a uniform acceleration along the
positive x axis. Identify all figure that correctly
4. All the graphs below are intended to represent represent the motion qualitatively, (
the same motion. One of them does it a=acceleration, v=velocity, x=displacement,
incorrectly. Pick it up. t=time)

5. A particle starts from the origin at time t=0 and

moves along the positive x- axis. The graph of 8. Consider an expanding sphere of
velocity with respect to time is shown in figure. instantaneous radius R whose total mass
What is the position of the particle at time t=5 remain constant. The expansion is such that
s? the instantaneous density p remains uniform
throughout the volume. The rate of fractional
change in density (1 dp/p dt) is constant. The
velocity v of any point on the surface of the
expanding sphere is proportional to
(a) R3
(b) R
(c) R 2/3
(d) 1/r

(a) 10 m
(b) 6m
(c) 3m
(d) 9m

6. A bullet of mass 20 g has an initial speed of 1

ms-1, just before it starts penetrating a mud
wall of thickness 20 cm, if the wall offers
mean resistance of 2.5×10-2 N, the speed of

45