Exercises

PROJECTILE
LEVEL 1
Assertion and Reason
Directions Choose the correct option.
(a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion.
(b) If both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
(c) If Assertion is true, but the Reason is false.
(d) If Assertion is false but the Reason is true.
1. Assertion : A particle follows only a parabolic path if acceleration is constant.
Reason : In projectile motion path is parabolic, as acceleration is assumed to be constant at
low heights.
2. Assertion : Projectile motion is called a two dimensional motion, although it takes place in
space.
Reason : In space it takes place in a plane.
3. Assertion : If time of flight in a projectile motion is made two times, its maximum height will
become four times.
Reason : In projectile motion H ∝ T 2, where H is maximum height and T the time of flight.
4. Assertion : A particle is projected with velocity u at angle 45° with ground. Let v be the
velocity of particle at time t ( ≠ 0), then value of u ⋅ v can be zero.
Reason : Value of dot product is zero when angle between two vectors is 90°.
5. Assertion : A particle has constant acceleration is x - y plane. But neither of its acceleration
components ( ax and a y ) is zero. Under this condition particle cannot have parabolic path.
Reason : In projectile motion, horizontal component of acceleration is zero.
v − v1
6. Assertion : In projectile motion at any two positions 2 always remains constant.
t2 − t1
Reason : The given quantity is average acceleration, which should remain constant as
acceleration is constant.
7. Assertion : Particle A is projected upwards. Simultaneously particle B is
projected as projectile as shown. Particle A returns to ground in 4 s. At the H
same time particle B collides with A. Maximum height H attained by B
would be 20 m. ( g = 10 ms−2 ) B A

Reason : Speed of projection of both the particles should be same under the given condition.
8. Assertion : Two projectiles have maximum heights 4H and H respectively. The ratio of their
horizontal components of velocities should be 1 : 2 for their horizontal ranges to be same.
Reason : Horizontal range = horizontal component of velocity × time of flight.

g

       
 



 ­€‚

 
 Chapter 7 Projectile Motion  247

9. Assertion : If g = 10 m/ s2 then in projectile motion speed of particle in every second will

change by 10 ms−1.
Reason : Acceleration is nothing but rate of change of velocity.
10. Assertion : In projectile motion if particle is projected with speed u, then speed of particle at
height h would be u 2 − 2gh .
Reason : If particle is projected with vertical component of velocity u y . Then vertical
component at the height h would be ± u 2y − 2gh

Objective Questions
Single Correct Option
1. Identify the correct statement related to the projectile motion.
(a) It is uniformly accelerated everywhere
(b) It is uniformly accelerated everywhere except at the highest position where it is moving with
constant velocity
(c) Acceleration is never perpendicular to velocity
(d) None of the above
2. Two bodies are thrown with the same initial velocity at angles θ and ( 90° − θ) respectively with
the horizontal, then their maximum heights are in the ratio
(a) 1 : 1 (b) sin θ : cos θ (c) sin 2 θ : cos 2 θ (d) cos θ : sin θ
3. The range of a projectile at an angle θ is equal to half of the maximum range if thrown at the
same speed. The angle of projection θ is given by
(a) 15° (b) 30° (c) 60° (d) data insufficient
4. A ball is projected with a velocity 20 ms−1 at an angle to the horizontal. In order to have the
maximum range. Its velocity at the highest position must be
(a) 10 ms −1 (b) 14 ms −1 (c) 18 ms −1 (d) 16 ms −1
^ ^ ^ ^
5. A particle has initial velocity, v = 3 i + 4 j and a constant force F = 4 i − 3 j acts on it. The path of
the particle is
(a) straight line (b) parabolic (c) circular (d) elliptical
6. A body is projected at an angle 60° with the horizontal with kinetic energy K. When the velocity
makes an angle 30° with the horizontal, the kinetic energy of the body will be
(a) K/2 (b) K/3 (c) 2 K/ 3 (d) 3 K/ 4
7. If T1 and T2 are the times of flight for two complementary angles, then the range of projectile R is
given by
1 1
(a) R = 4 gT1T2 (b) R = 2 gT1T2 (c) R = gT1T2 (d) R = gT1T2
4 2
8. A gun is firing bullets with velocity v0 by rotating it through 360° in the horizontal plane. The
maximum area covered by the bullets is
πv02 π 2v02 πv04 π 2v04
(a) (b) (c) (d)
g g g2 g
9. A grass hopper can jump maximum distance 1.6 m. It spends negligible time on ground. How
far can it go in 10 2 s ?
(a) 45 m (b) 30 m (c) 20 m (d) 40 m

g

       
 



 ­€‚

 
 248  Mechanics - I

10. Two stones are projected with the same speed but making different angles with the horizontal.
π
Their horizontal ranges are equal. The angle of projection of one is
and the maximum height
3
reached by it is 102 m. Then the maximum height reached by the other in metres is
(a) 76 (b) 84
(c) 56 (d) 34

11. A ball is projected upwards from the top of a tower with a velocity 50 ms−1 making an angle 30°
with the horizontal. The height of tower is 70 m. After how many seconds from the instant of
throwing, will the ball reach the ground. ( g = 10 ms−2 )
(a) 2 s (b) 5 s (c) 7 s (d) 9 s
12. Average velocity of a particle in projectile motion between its starting point and the highest
point of its trajectory is (projection speed = u, angle of projection from horizontal = θ)
u
(a) u cos θ (b) 1 + 3 cos 2 θ
2
u u
(c) 2 + cos 2 θ (d) 1 + cos 2 θ
2 2
13. A train is moving on a track at 30 ms−1. A ball is thrown from it perpendicular to the direction of
motion with 30 ms−1 at 45° from horizontal. Find the distance of ball from the point of projection
on train to the point where it strikes the ground.
(a) 90 m (b) 90 3 m (c) 60 m (d) 60 3 m

14. A body is projected at time t = 0 from a certain point on a planet’s surface with a certain velocity
at a certain angle with the planet’s surface (assumed horizontal). The horizontal and vertical
displacements x and y (in metre) respectively vary with time t in second as, x = (10 3 ) t and
y = 10 t − t 2. The maximum height attained by the body is
(a) 75 m (b) 100 m
(c) 50 m (d) 25 m

15. A particle is fired horizontally from an inclined plane of inclination 30° with horizontal with
speed 50 ms−1. If g = 10 ms−2, the range measured along the incline is
1000
(a) 500 m (b) m (c) 200 2 m (d) 100 3 m
3
16. A fixed mortar fires a bomb at an angle of 53° above the horizontal with a muzzle velocity of
80 ms−1. A tank is advancing directly towards the mortar on level ground at a constant speed of
5 m/s. The initial separation (at the instant mortar is fired) between the mortar and tank, so
that the tank would be hit is [ Take g = 10 ms−2 ]
(a) 662.4 m (b) 526.3 m
(c) 486.6 m (d) None of these

Subjective Questions
1. At time t = 0, a small ball is projected from point A with a velocity of 60 m/s at 60° angle with
horizontal. Neglect atmospheric resistance and determine the two times t1 and t2 when the
velocity of the ball makes an angle of 45° with the horizontal x-axis.
2. A particle is projected from ground with velocity 20 2 m/s at 45°. At what time particle is at
height 15 m from ground? ( g = 10 m/ s2 )

g

       
 



 ­€‚

 
 Chapter 7 Projectile Motion  249

3. A particle is projected at an angle 60° with horizontal with a speed v = 20 m/s. Taking
g = 10 m/ s2. Find the time after which the speed of the particle remains half of its initial speed.
4. Two particles A and B are projected from ground towards each other with speeds 10 m/s and
5 2 m/s at angles 30° and 45° with horizontal from two points separated by a distance of 15 m.
Will they collide or not?
5√2 m/s
10 m/s

30° 45°
A B
15 m

5. Two particles move in a uniform gravitational field with an acceleration g. At the initial
moment the particles were located over a tower at one point and moved with velocities
v1 = 3 m/ s and v2 = 4 m/ s horizontally in opposite directions. Find the distance between the
particles at the moment when their velocity vectors become mutually perpendicular.
6. A ball is thrown from the ground to clear a wall 3 m high at a distance of 6 m and falls 18 m
away from the wall. Find the angle of projection of ball.
7. A body is projected up such that its position vector varies with time as r = { 3 t i + ( 4 t − 5 t 2 )j} m.
Here, t is in seconds. Find the time and x-coordinate of particle when its y-coordinate is zero.
8. A particle is projected along an inclined plane as shown in figure. What is the speed of the
particle when it collides at point A ? ( g = 10 m/ s2 )

u = 10 m/s
A

30°
30°
O

9. In the above problem, what is the component of its velocity perpendicular to the plane when it
strikes at A ?
10. Two particles A and B are projected simultaneously from two towers of heights 10 m and 20 m
respectively. Particle A is projected with an initial speed of 10 2 m/s at an angle of 45° with
horizontal, while particle B is projected horizontally with speed 10 m/s. If they collide in air,
what is the distance d between the towers?
10 m/s
B
10√2 m/s
45°
A
20 m
10m

g

       
 



 ­€‚

 
 250  Mechanics - I

11. A particle is projected from the bottom of an inclined plane of inclination 30° with velocity of
40 m/s at an angle of 60° with horizontal. Find the speed of the particle when its velocity vector
is parallel to the plane. Take g = 10 m/ s2.
12. Two particles A and B are projected simultaneously in the directions shown in figure with
1
velocities vA = 20 m/ s and vB = 10 m/ s respectively. They collide in air after s. Find
2
(a) the angle θ
(b) the distance x.
vB = 10 m/s
vA = 20 m/s

θ
A B
x

13. A ball is shot from the ground into the air. At a height of 9.1 m, its velocity is observed to be
v = 7.6i + 6.1j in metre per second (i is horizontal, jis upward). Give the approximate answers.
(a) To what maximum height does the ball rise?
(b) What total horizontal distance does the ball travel?
(c) What are the magnitude and
(d) What are the direction of the ball’s velocity just before it hits the ground?
14. A particle is projected with velocity 2 gh, so that it just clears two walls of equal height h which
h
are at a distance of 2h from each other. Show that the time of passing between the walls is 2 .
g
[Hint : First find velocity at height h. Treat it as initial velocity and 2h as the range.]
15. A particle is projected at an angle of elevation α and after t second it appears to have an
elevation of β as seen from the point of projection. Find the initial velocity of projection.
16. A projectile aimed at a mark, which is in the horizontal plane through the point of projection,
falls a cm short of it when the elevation is α and goes b cm far when the elevation is β.Show that,
if the speed of projection is same in all the cases the proper elevation is
1  b sin 2α + a sin 2β 
sin−1  
2  a+b 
17. Two particles are simultaneously thrown in horizontal direction from two points on a
riverbank, which are at certain height above the water surface. The initial velocities of the
particles are v1 = 5 m/ s and v2 = 7.5 m/ s respectively. Both particles fall into the water at the
same time. First particle enters the water at a point s = 10 m from the bank. Determine
(a) the time of flight of the two particles,
(b) the height from which they are thrown,
(c) the point where the second particle falls in water.
18. A balloon is ascending at the rate v = 12 km/ h and is being carried horizontally by the wind at
vw = 20 km/ h.If a ballast bag is dropped from the balloon at the instant h = 50 m, determine the
time needed for it to strike the ground. Assume that the bag was released from the balloon with
the same velocity as the balloon. Also, find the speed with which the bag strikes the ground?

g

       
 



 ­€‚

 
 Chapter 7 Projectile Motion  251

19. A projectile is fired with a velocity u at right angles to the slope, which is inclined at an angle θ
with the horizontal. Derive an expression for the distance R to the point of impact.

θ .

20. An elevator is going up with an upward acceleration of 1 m/ s2. At the instant when its velocity
is 2 m/s, a stone is projected upward from its floor with a speed of 2 m/s relative to the elevator,
at an elevation of 30°.
(a) Calculate the time taken by the stone to return to the floor.
(b) Sketch the path of the projectile as observed by an observer outside the elevator.
(c) If the elevator was moving with a downward acceleration equal to g, how would the motion be
altered?
21. Two particles A and B are projected simultaneously in a vertical plane as shown in figure. They
collide at time t in air. Write down two necessary equations for collision to take place.
y (m)
u2

θ2
20 B
u1
θ1
10
A

x (m)
10 30

LEVEL 2
Objective Questions
Single Correct Option
1. Two bodies were thrown simultaneously from the same point, one straight up, and the other, at
an angle of θ = 30° to the horizontal. The initial velocity of each body is 20 ms−1. Neglecting air
resistance, the distance between the bodies at t = 1.2 later is
(a) 20 m (b) 30 m
(c) 24 m (d) 50 m
2. A particle is dropped from a height h. Another particle which is initially at a horizontal distance
d from the first is simultaneously projected with a horizontal velocity u and the two particles
just collide on the ground. Then
u 2h 2u 2h
(a) d 2 = (b) d 2 =
2h g
(c) d = h (d) gd 2 = u 2h

g

       
 



 ­€‚

 
 252  Mechanics - I

3. A ball is projected from point A with velocity 10 ms−1perpendicular to the

90°
inclined plane as shown in figure. Range of the ball on the inclined plane is A
40 20 12 60
(a) m (b) m (c) m (d) m
3 3 3 3 30°
4. A heavy particle is projected with a velocity at an angle with the horizontal
into the uniform gravitational field. The slope of the trajectory of the particle varies as

slope

slope
slope

slope
(a) O t (b) O x (c) O t (d) O x

5. A particle starts from the origin of coordinates at time t = 0 and moves in the xy plane with a
constant acceleration α in the y-direction. Its equation of motion is y = βx 2. Its velocity
component in the x-direction is
2α α α
(a) variable (b) (c) (d)
β 2β 2β
6. A projectile is projected with speed u at an angle of 60° with horizontal from the foot of an
inclined plane. If the projectile hits the inclined plane horizontally, the range on inclined plane
will be
u 2 21 3u 2 u2 21 u 2
(a) (b) (c) (d)
2g 4g 2β 8g

7. A particle is projected at an angle 60° with speed 10 3 m/s, from the

10√3 m/s 10 3 m/s
point A, as shown in the figure. At the same time the wedge is made √
to move with speed 10 3 m/s towards right as shown in the figure. 30° 60°
Then the time after which particle will strike with wedge is
4
(a) 2 s (b) 2 3 s (c) s (d) None of these
3
8. A particle moves along the parabolic path x = y 2 + 2 y + 2 in such a way that Y -component of
velocity vector remains 5 ms−1 during the motion. The magnitude of the acceleration of the
particle is
(a) 50 ms −2 (b) 100 ms −2 (c) 10 2 ms −2 (d) 0.1 ms −2
9. A shell fired from the base of a mountain just clears it. If α is the angle of
projection, then the angular elevation of the summit β is
α  1
(a) (b) tan −1  
2  2 H
β
 tan α 
(c) tan −1   (d) tan −1 (2 tan α )
 2 

10. In the figure shown, the two projectiles are fired simultaneously. The 20 √ 3 m/s
minimum distance between them during their flight is 20 m/s
(a) 20 m
(b) 10 3 m
60° 30°
(c) 10 m
20 √ 3 m
(d) None of the above

g

       
 



 ­€‚

 
 Chapter 7 Projectile Motion  253

More than One Correct Options

1. Two particles projected from the same point with same speed u at angles of projection α and β
strike the horizontal ground at the same point. If h1 and h2 are the maximum heights attained
by the projectile, R is the range for both and t1 and t2 are their times of flights, respectively ,
then
π t1 h1
(a) α + β = (b) R = 4 h1h2 (c) = tan α (d) tan α =
2 t2 h2

2. A ball is dropped from a height of 49 m. The wind is blowing horizontally. Due to wind a
constant horizontal acceleration is provided to the ball. Choose the correct statement (s).
[Take g = 9.8 ms−2]
(a) Path of the ball is a straight line
(b) Path of the ball is a curved one
(c) The time taken by the ball to reach the ground is 3.16 s
(d) Actual distance travelled by the ball is more then 49 m
3. A particle is projected from a point P with a velocity v at an angle θ with horizontal. At a certain
point Q it moves at right angles to its initial direction. Then
(a) velocity of particle at Q is v sin θ
(b) velocity of particle at Q is v cot θ
(c) time of flight from P to Q is (v/g ) cosecθ
(d) time of flight from P to Q is (v/g ) sec θ
4. At a height of 15 m from ground velocity of a projectile is v = (10 i + 10j). Here, j is vertically
upwards and i is along horizontal direction then ( g = 10 ms−2 )
(a) particle was projected at an angle of 45° with horizontal
(b) time of flight of projectile is 4 s
(c) horizontal range of projectile is 100 m
(d) maximum height of projectile from ground is 20 m
5. Which of the following quantities remain constant during projectile motion?
(a) Average velocity between two points (b) Average speed between two points
dv d 2v
(c) (d) 2
dt dt
6. In the projectile motion shown is figure, given tAB = 2 s then ( g = 10 ms−2 )

A B

15 m
O
20 m 40 m B

(a) particle is at point B at 3 s

(b) maximum height of projectile is 20 m
(c) initial vertical component of velocity is 20 ms −1
(d) horizontal component of velocity is 20 ms −1

g

       
 



 ­€‚

 
 254  Mechanics - I

Comprehension Based Questions

Passage (Q. Nos. 1 to 2)
Two inclined planes OA and OB intersect in a horizontal plane u
having their inclinations α and β with the horizontal as shown in A B
figure. A particle is projected from point P with velocity u along a P a Q
direction perpendicular to plane OA. The particle strikes plane OB α β
perpendicularly at Q. O
1. If α = 30° , β = 30°, the time of flight from P to Q is
u 3u 2u 2u
(a) (b) (c) (d)
g g g g
2. If α = 30° , β = 30° and a = 4.9 m, the initial velocity of projection is
(a) 9.8 ms −1 (b) 4.9 ms −1 (c) 4.9 2 ms −1 (d) 19.6 ms −1
Match the Columns
1. Particle-1 is just dropped from a tower. 1 s later particle-2 is thrown from the same tower
horizontally with velocity 10 ms−1. Taking g = 10 ms−2, match the following two columns at
t = 2 s.
Column I Column II
(a) Horizontal displacement between two (p) 10 SI units
(b) Vertical displacement between two (q) 20 SI units
(c) Magnitude of relative horizontal component of velocity (r) 10 2 SI units
(d) Magnitude of relative vertical component of velocity (s) None of the above

R
2. In a projectile motion, given H = = 20 m. Here, H is maximum height and R the horizontal
2
range. For the given condition match the following two columns.
Column I Column II
(a) Time of flight (p) 1
(b) Ratio of vertical component of velocity and horizontal (q) 2
component of velocity
(c) Horizontal component of velocity (in m/s) (r) 10
(d) Vertical component of velocity (in m/s) (s) None of the above

3. A particle can be thrown at a constant speed at different angles. When it is thrown at 15° with
horizontal, it falls at a distance of 10 m from point of projection. For this speed of particle match
following two columns.
Column I Column II
(a) Maximum horizontal range which can be taken with (p) 10 m
this speed
(b) Maximum height which can be taken with this speed (q) 20 m
(c) Range at 75° (r) 15 m
(d) Height at 30° (s) None of the above

g

       
 



 ­€‚

 
 Chapter 7 Projectile Motion  255

4. In projectile motion, if vertical component of velocity is increased to two times, keeping

horizontal component unchanged, then
Column I Column II
(a) Time of flight (p) will remain same
(b) Maximum height (q) will become two times
(c) Horizontal range (r) will become four times
(d) Angle of projection with (s) None of the above
horizontal

5. In projectile motion shown in figure.

A
u

θ
O B

Column I Column II
(a) Change in velocity between O and A (p) u cos θ
(b) Average velocity between O and A (q) u sin θ
(c) Change in velocity between O and B (r) 2 u sin θ
(d) Average velocity between O and B (s) None of the above

6. Particle-1 is projected from ground (take it origin) at time t = 0, with velocity ( 30i + 30j) ms−1.
Particle-2 is projected from (130 m, 75 m) at time t = 1 s with velocity ( −20 i + 20 j) ms−1.
Assuming j to be vertically upward and i to be in horizontal direction, match the following two
columns at t = 2 s.
Column I Column II
(a) horizontal distance between two (p) 30 SI units
(b) vertical distance between two (q) 40 SI units
(c) relative horizontal component of velocity between two (r) 50 SI units
(d) relative vertical component of velocity between two (s) None of the above

7. The trajectories of the motion of three particles are shown in the figure. Match the entries of
Column I with the entries of Column II. Neglect air resistance.
y

A B C
x

g

       
 



 ­€‚

 
 256  Mechanics - I

Column I Column II
(a) Time of flight is least for (p) A
(b) Vertical component of velocity is greatest for (q) B
(c) Horizontal component of velocity is greatest for (r) C
(d) Launch speed is least for (s) same for all

Subjective Questions
1. Determine the horizontal velocity v0 with which a stone must be projected horizontally from a
point P, so that it may hit the inclined plane perpendicularly. The inclination of the plane with
the horizontal is θ and point P is at a height h above the foot of the incline, as shown in the
figure.
v0 P

2. A particle is dropped from point P at time t = 0. At the same time another particle is thrown
from point O as shown in the figure and it collides with the particle P. Acceleration due to
gravity is along the negative y-axis. If the two particles collide 2 s after they start, find the
initial velocity v0 of the particle which was projected from O. Point O is not necessarily on
ground.
y 2m P

10 m
v0

θ
O x

3. Two particles are simultaneously projected in the same vertical plane from the same point with
velocities u and v at angles α and β with horizontal. Find the time that elapses when their
velocities are parallel.
4. A projectile takes off with an initial velocity of 10 m/s at an angle of elevation of 45°. It is just
able to clear two hurdles of height 2 m each, separated from each other by a distance d.
Calculate d. At what distance from the point of projection is the first hurdle placed? Take
g = 10 m/ s2.

5. A stone is projected from the ground in such a direction so as to hit a bird on the top of a
telegraph post of height h and attains the maximum height of 2h above the ground. If at the
instant of projection, the bird were to fly away horizontally with a uniform speed, find the ratio
between the horizontal velocity of bird and the horizontal component of velocity of stone, if the
stone hits the bird while descending.

g

       
 



 ­€‚

 
 Chapter 7 Projectile Motion  257

6. A particle is released from a certain height H = 400 m. Due to the wind, the particle gathers the
horizontal velocity component vx = ay where a = 5 s−1 and y is the vertical displacement of the
particle from the point of release, then find
(a) the horizontal drift of the particle when it strikes the ground,
(b) the speed with which particle strikes the ground.
(Take g = 10 m/s 2)
7. A train is moving with a constant speed of 10 m/s in a circle of radius y
16
m. The plane of the circle lies in horizontal x-y plane. At time t = 0,
π
train is at point P and moving in counter-clockwise direction. At this
instant, a stone is thrown from the train with speed 10 m/s relative to
train towards negative x-axis at an angle of 37° with vertical z-axis.
Find P x
(a) the velocity of particle relative to train at the highest point of its
trajectory.
(b) the co-ordinates of points on the ground where it finally falls and that
of the highest point of its trajectory.
3
Take g = 10 m/s 2, sin 37° =
5
8. A particle is projected from an inclined plane OP1 from A with velocity v1 = 8 ms−1 at an
angle 60° with horizontal. An another particle is projected at the same instant from B with
velocity v2 = 16 ms−1 and perpendicular to the plane OP2 as shown in figure. After time
10 3 s there separation was minimum and found to be 70 m. Then find distance AB.
P1 v1 v2
P2
60°
A
90°
B
45° 30°
O

9. A particle is projected from point O on the ground with velocity Y

u = 5 5 m/s at angle α = tan−1 ( 0.5 ). It strikes at a point C on a
fixed smooth plane AB having inclination of 37° with horizontal as 5√5 m/s B
C
shown in figure. If the particle does not rebound, calculate
y
(a) coordinates of point C in reference to coordinate system as shown α 37°
in the figure. O
A D
X
(10/3) m
(b) maximum height from the ground to which the particle rises.
( g = 10 m/s 2).
10. A plank fitted with a gun is moving on a horizontal surface with speed of 4 m/s along the
positive x-axis. The z-axis is in vertically upward direction. The mass of the plank including the
mass of the gun is 50 kg. When the plank reaches the origin, a shell of mass 10 kg is fired at an
angle of 60° with the positive x-axis with a speed of v = 20 m/s with respect to the gun in
x-z plane. Find the position vector of the shell at t = 2 s after firing it. Take g = 9.8 m/ s2.

g

       
 



 ­€‚

 
 3. 2.31 s, 53.33 mm
4. (a) A vertical straight line (b) A parabola
5. (a) zero (b) 20 ms −1 in horizontal direction (c) 40 m
6. 60°

Exercises
LEVEL 1
Assertion and Reason
1. (d) 2. (a) 3. (a) 4. (b) 5. (d) 6. (a) 7. (c) 8. (a or b) 9. (d) 10. (b)

Single Correct Option

1. (a) 2. (c) 3. (a) 4. (b) 5. (b) 6. (b) 7. (d) 8. (c) 9. (d) 10. (d)
11. (c) 12. (b) 13. (a) 14. (d) 15. (b) 16. (d)
Chapter 7 Projectile Motion  259

Subjective Questions
1 t1 = 2.19s, t 2 = 8.20 s 2. 3 s and 1 s 3. 3 s

6. tan−1  
2
4. No 5. 2.5 m
 3
10
7. time = zero, 0.8 s,x-coordinate = 0, 2.4 m 8. m/s 9. 5 m/s
3
40
10. 20 m 11. m/s 12. (a) 30° (b) 5 3 m
3
13. (a) 11 m, (b) 23 m (c) 16.6 m/s (d) tan−1 (2), below horizontal
gt cos β
15. u = 17. (a) 2s (b) 19.6 m (c) 15 m
sin (α − β )
2 u2
18. 3.55 s, 32.7 m/s 19. R = tan θ sec θ
g
20. (a) 0.18 s (c) a straight line with respect to elevator and projectile with respect to ground
21. (u1 cos θ1 + u2 cos θ 2 ) t = 20 ...(i) (u1 sin θ1 − u2 sin θ 2 ) t = 10 ...(ii)

LEVEL 2
Single Correct Option
1. (c) 2. (b) 3. (a) 4. (a) 5. (d) 6. (d) 7. (a) 8. (a) 9. (c) 10. (b)

More than One Correct Options

1. (all) 2. (a,c,d) 3. (b,c) 4. (b,d) 5. (c,d) 6. (all)

Comprehension Based Questions

1. (b) 2. (a)

Match the Columns

1. (a) →(p), (b) →(s), (c) →(p), (d) →(p) 2. (a) →(s), (b) →(q), (c) →(r), (d) →(s)
3. (a) →(q), (b) →(p), (c) →(p), (d) →(s) 4. (a) →(q), (b) →(r), (c) →(q), (d) →(s)
5. (a) →(q), (b) →(s), (c) →(r), (d) →(p) 6. (a) →(r), (b) →(r), (c) →(r), (d) →(s)
7. (a) →(s), (b) →(s), (c) →(r), (d) →(p)

Subjective Questions
2 gh
1. v0 = 2. 26 ms −1 at angle θ = tan−1 (5) with x-axis

g
2 + cot2 θ


uv sin (α − β )
3. t = 4. 4.47
 m,
 2.75 m
g (v cos β 
− u cos α )    
   



 ­€‚

2

5. 6. (a) 2.67 km (b) 0.9 km/s
2+1
 2 gh
1. v0 = 2. 26 ms −1 at angle θ = tan−1 (5) with x-axis
2 + cot2 θ
uv sin (α − β )
3. t = 4. 4.47 m, 2.75 m
g (v cos β − u cos α )
2
5. 6. (a) 2.67 km (b) 0.9 km/s
2+1

7. (a) (−6 i + 10 j ) ms −1 (b) (−4.5 m, 16 m, 0), (0.3 m, 8.0 m, 3.2 m)
8. 250 m 9. (a) (5 m, 1.25 m) (b) 4.45 m
10. [ 24 i + 15 k
] m

g

       
 



 ­€‚