1. A car is driven over a measured 2 km in 1.5 min.
Determine the speed of the car (a) in meters per
second and (b) in feet per second.
2. A river steamer can travel at the rate of 24 km/hr in still water. How long will the trip between two
cities 95 km apart take (a) downstream and (b) upstream, if the river current is 4.5 km/hr? (c) What will
be the average speed for the round trip?
3. A ferryboat which can sail at the rate of 8 mi/hr in still water travels straight across a river t mi wide in
which there is a current of 2 mi/hr. (a) What is the velocity of the ferryboat with respect to the shore? (b)
How long does a trip take?
4. The first three runners in a 100-m race were clocked in 9.5 sec, 10.0 sec, and 10.5 sec, respectively. (a)
What was the average speed of each runner and (b) how far apart were the first and last runners when the
first one reached the finish line?
5. An airplane heads due north with a velocity of 400 km/hr. A west wind is blowing with a velocity of 60
km/hr. What is the velocity of the airplane relative to the ground?
6. An airplane whose normal speed in still air is 415 km/hr must travel due east. (a) What course must the
aviator set for the plane when there is a steady southwest wind of 6o km/hr? (b) How long will it take to
travel 1400 km?
7. An automobile starting from rest acquires a speed of 60 km/hr in 12 sec. What is its average
acceleration?
8. How long in seconds will it take for a car, starting from rest, to acquire a speed of 95 km/hr if its
acceleration is 12 ft/sec²?
9. The brakes are applied to the wheels of a locomotive when it is traveling at 110 km/hr. It comes to rest
24 sec after the brakes are applied. What is its average acceleration in m/s²?
10. An automobile which is traveling at a speed of 55 mi/hr must be brought to a stop within 150 ft. What is
the minimum acceleration that must be given to the car to accomplish this?
11. An airplane taking off on a runway 365 m long must acquire a speed of 130 km/hr to get safely into the
air. (a) What is the minimum safe acceleration for this airplane in m/s²? (b) How long will it take for the
airplane to acquire this speed when so accelerated in seconds?
12. A car approaching a turn in the road has its speed decreased from 50 mi/hr to 30 mi/hr while traversing a
distance of 120 ft. (a) What was its acceleration and (b) how long did it take to traverse this distance?
13. A boy drops a stone from a bridge 80 ft above the water. (a) With what speed did the stone strike the
water? (b) With what speed would the stone have struck the water if it had been thrown down with a
speed of 24 ft/sec?
14. A boy throws a ball vertically upward and catches it 1 sec later. (a) How high up did the ball go? (b)
With what speed was it thrown upward?
15. A boy throws a stone horizontally with a speed of 30 ft/sec from a cliff 256 ft high. (a) How long will it
take the stone to strike the ground? (b) Where will the stone land? (c) With what velocity will the stone
strike the ground?
16. A small block starting from rest takes 5 sec to slide down an inclined plane 80 cm long. (a) What was
its acceleration and (b) with what speed did it reach the bottom of the incline?
17. Two horizontal wires are placed parallel to each other 100 cm apart, one directly above the other. A
falling ball is clocked as it passes each of these wires. If the time elapsed is 0.20 sec, determine the
speed the ball had when it passed each wire.
18. Fighter planes fly at 35,000 ft elevation. What must be the muzzle velocity of an anti-aircraft shell to
reach this height neglecting air resistance?
19. A rifle fires a bullet with a speed of 30,000 cm/sec. If the elevation of the rifle is 30° with the horizontal,
determine (a) the range of the bullet on horizontal ground and (b) the velocity of the bullet when it
reaches the ground.
20. A projectile is fired vertically upward with an initial velocity of 1,800 ft/sec. (a) How high does it rise?
(b) What velocity will it have 5 sec after leaving the gun? (c) What is its altitude 5 sec after leaving the
gun?
21. A car moving with a speed of 30 mi/hr reaches the top of a hill. As it goes down the hill, its speed
increases to 45 mi/hr in 1.5 min. (a) What is the acceleration of the car and (b) what distance does it
travel in this time?
22. A stone thrown horizontally from a hill takes 6 sec to reach the ground. Determine, in meters, the height
of the hill.
23. A falling stone is seen to pass a window 2 m high in 0.3 sec. (a) Determine the average speed of the
stone. (b) Determine the speed with which it reaches the level of the top of the window. (c) Determine
the height above this point from which it fell.
24. The distance between two stop lights on a cross-town street is 800 ft. If the acceleration of a certain car,
both positive and negative, is kept at 6 ft/sec² , and if the speed limit on this street is 30 mi/hr, determine
the minimum time to traverse this distance.
25. A ball is thrown a distance of 65 ft in 1.2 sec. Assuming that it was caught at the same level as it was
thrown, (a) determine how high the ball rose in its path of motion. (b) With what velocity was the ball
thrown?
26. A gun fires a shell with a velocity of 600 m/sec at an angle of 45° with the horizontal. Neglecting ail'
resistance, (a) determine the range of this gun, (b) determine the maximum height reached by the shell,
and (c) determine the time of flight of this shell on level ground.
27. Derive the equation for the range of a projectile fired on level ground, R = (u² sin 2θ)/g , where R is the
range, θ is the angle of elevation, and u is the initial g velocity. Show that the maximum range is
achieved when θ = 45°.
28. The x coordinate of an object moving along the x axis is given by the equation x = 3 - 5t + 12t² ft. Find
the corresponding equation for the velocity and acceleration of the object at any time t.
29. A body moving in space has its motion described by the equations x = 12t + 15, y = 6t² where the
distances are in meters and the time is given in seconds. Find the magnitude and direction of the velocity
and the acceleration when t = 3 sec.
30. A ball is thrown toward a building 50 ft distant at a speed of 100 ft/sec. At what angle must it be thrown
if it is to pass through a window 42 ft from the ground?
31. A railroad caris moving due north at a speed of 60 mi/hr. A ball is thrown from the window due east at
an angle of elevation of 30° and a speed of 40 ft/sec. (a) Find the time at which it strikes the ground 10 it
below the window of the car. (b) How far east of the track does the ball land? (c) How far north of the
point of projection does the ball land?
32. When a balloon is at a height of 6400 ft and rising at a speed of 32 ft/sec, a stone is thrown vertically out
of the balloon. The stone hits the ground directly below in 20 sec. (a) What was the initial velocity of the
stone relative to the balloon? (b) Relative to the ground?
33. Motorist A, starting from rest, accelerates at a rate of 6 ft/sec². At the same time that A begins, motorist
B, starting from rest at a point 100 ft ahead of A, accelerates at a rate of 4 ft/sec². (a) How far does
motorist A travel before they meet? (b) At the instant they meet each motorist decelerates at the rate of 5
ft/sec² until his car comes to rest. How far apart are they when they have stopped?
34. A train is moving with uniform speed along a level road. A man on the observation platform drops a
ball. What is the path of the ball as observed (a) by the man on the train and (b) by another person
standing at a short distance from the tracks?
35. In a laboratory experiment an air rifle is clamped in position and aimed by sighting along the barrel. The
target is released just as the bullet leaves the muzzle of the rifle. Show that the bullet will always hit the
target.
36. If there is no wind, raindrops fall vertically with uniform speed. A man driving a car on a windless rainy
day observes that the tracks left by the raindrops on the side windows are all inclined at the same angle.
Show how the vertical speed of the raindrops can be determined from the inclination of the tracks and
the reading of the speedometer.
37. Show that the speeds of a projectile are the same at any two points in its path which are at the same
elevation.
38. A boy seated in a rapidly moving railroad car tosses a ball up into the air. Will the ball come down in
front of him; behind him; into his hand? What will happen when the car is accelerating in the forward
direction? Going round a curve?
Rectilinear Kinematics
39. A stone is dropped into a well and the splash is heard two seconds later. If sound travels 1200 km/hr,
what is the depth of the well in meters? H = 60.6 ft
40. In the two pulley systems shown in the figure shown, determine the velocity and acceleration of block 3
when blocks 1 and 2 have the velocities and acceleration shown. v3 = 1 ft/sec; a3 = 0
41. A particle moves from point A to B under the influence of a variable force. It has an acceleration of 5
ft/s2 for 6 s, and then its acceleration increases to 7 ft/s 2 until its has reached the velocity of 100 ft/s. For
sometime its velocity remains constant, then the particle starts to decelerate and within 10 s stops at B.
The total travel time from A to B is 46 s. Draw the a-t, v-t, and x-t curves, and determine the distance
between points A and B. x (t=46 sec) = 3240 ft
a-t curve v-t curve x-t curve
42. Two loads A and B are carried by a system of three pulleys C, D, E as shown Fig. 1. Pulley D is moving
downward with a constant velocity of 2 m/s, pulleys C and E are fixed. At a moment t = 0 load A starts
to move (initial velocity v = 0) downward from the position M with the constant acceleration. Load A
has velocity 8 m/s when it passes through N, MN = 4m. Calculate the change in elevation, the velocity
and the acceleration of load B when A passes through N.
43. The motion of a particle along a straight line is described by the equation x = t 3 − 3t2 − 45t + 50, where
x is expressed in feet and t in seconds. Compute a) the time at which v (t) = 0, b) the position and
distance traveled by the particle at that time, c) the acceleration of the particle at that time, d) the
distance traveled by the particle from t = 4 sec. to t = 6 sec.
Projectile Motion
44. The total speed of a projectile at its greatest height, v 1, is √(6/7) of its total speed when it is at half its
greatest height, v2 . Show that the angle of projection is 30°.
45. A projectile, fired with velocity v0 and angle α0 from the horizontal, is to impact at point P as shown
below. What is the range measured along the straight line connecting the firing point and the target?
46. An airplane is traveling horizontally at 480 mph at a height of 6400 ft. The airplane drops a bomb aimed
at a stationary target on the ground. To an observer on the aircraft, what angle must the target make with
the vertical, when the bomb is dropped, for the bomb to hit the target? Neglect air resistance. (See the
figure.) Suppose that the target is a ship which is steaming at 20 mph away from the aircraft along its
line of flight. What alterations would need to be made to the previous calculations?
47. A gun is fired at a moving target. The bullet has a projectile speed of 1200 ft/sec. Both gun and target are
on level ground. The target is moving 60 mph away from the gun and it is 30,000 ft from the gun at the
moment of firing. Determine the angle of elevation θ needed for a direct hit.
48. A stone is thrown from the top of a 200 m building with an initial velocity of 150 m/s at an angle of 30°
with the horizontal line. Neglecting the air resistance, determine a) the horizontal distance from the
building to the point where the stone lands, b) the maximum height above the ground reached by the
stone.
49. A rocket is launched from point B and its flight is tracked by an optical instrument from point A.
Express the velocity in terms of s, θ and θ ̇.
50. In a gymnasium with a ceiling 30 ft high, a player throws a ball towards a wall 80 ft away. If he releases
the ball 5 ft above the floor with initial velocity v0 of 55 ft/sec, determine the highest point at which the
ball could strike the wall.
51. A projectile of relatively small dimensions is fired with a velocity v 0 from a gun at an angle of elevation
θ with the horizontal, as shown in the figure. Neglecting air resistance, determine the maximum height h
and the distance r from the gun to the impact point P if we assume the gun is aimed down a hill whose
surface has an angle β with the horizontal.
52. A mortar at point A must lob shells over the cliff shown, (a) How far back must it be placed to shoot
cleanly over the cliff? (b) How far beyond the edge will the shell strike?
Equations of Motion in Two Dimensions
53. A race car is driving at a speed of 80 mi/h along a curved road of radius 2000 ft. The brakes are applied
causing constant deceleration of the car, after 8s the speed decreases to 50 mi/h. Calculate the
acceleration of the car at the instant after the brakes have been applied.
54. Assume a particle moves in the x−y plane such that its x and y coordinates at any time t are governed by
ẍ − 2bẏ + λx = 0 (a)
ӱ + 2bẋ + λy = 0. (b)
Find the equation of the motion.
55. A particle moves along the path r = 3φ so that φ = 2t3. Time is in seconds, φ is in radians, and r is in feet.
Determine the velocity of the particle when φ = 0.5 rad.
56. Determine the acceleration of the particle from the previous problem when φ = 0.5 rad.
57. A particle moves along the curve y = x3/2 such that its distance from the origin, measured along the
curve, is given by s = t3. Determine the acceleration when t = 2 sec. Units used are inches and seconds.
58. A particle moves along the path y = x2 in such a way that x = 2t. Determine the velocity and the
acceleration in terms of time t.
59. A body moves on the curve illustrated in the figure from point A to point B with a constant acceleration.
The velocity of the particle at A is 10 ft/sec and 10 seconds later at B it is 50 ft/sec. What is the total
acceleration of the particle at point B?
60. An airplane is attempting to fly with a constant speed, v, from point A ≡ (a, 0), on the x axis of the
accompanying figure, to the origin, 0. A wind blowing with speed w in the positive y direction will
greatly affect the flight of the plane. The pilot, who is not familiar with vectors, always points his plane
toward 0, thinking this to be the shortest way. Find the path that he is actually taking due to his mistake.
Newton's Second Law of Motion
61. If the pulleys have negligible mass and there is no friction, show that: (a) the acceleration of the blocks
A and B are g/7 and 2g/7; (b) the tension in the string is 5.71 lbf .
62. In the modified Atwood machine shown, both pulleys have negligible rotational inertia but the movable
pulley does have mass m. Suppose the mass m 3 is large enough to give a downward acceleration a' . Find
the acceleration a, m1 > m2, in terms of m1, m2, a' and g.
63. A body A on a horizontal frictionless table is connected to a string which runs over a pulley and carries a
platform on its other end, as shown in Fig. 1. A body C is placed on the platform. The weights of A, B,
and C are 10 lb, 2 lb, and 3 lb, respectively. What is the acceleration of A when the system is released
from rest, and what is the tension S in the string? Determine the contact force between B and C.
64. The system of two blocks A and B and two pulleys C and D is assembled as shown (Fig. 1). Neglecting
the friction and the mass of the pulleys and assuming that the whole system is initially at rest, determine
the acceleration of each block and the tension in each cord.
0.50 kN
2 kN
65. Two springs S1 and S2 of equal lengths L = 0.5 m, but with different spring constants K 1 = 50n/m and
K2 = 100 n/m, are joined and fastened between two supports A and B which are a distance 2L apart, as
shown in Fig.1. A body C of mass m = 2.5 kgm is fastened to the springs at their junction and is pulled
downward vertically until the length of each spring has doubled. The body is then released. What is the
initial acceleration of the body?
66. Two springs, S1 and S2, of negligible mass, with spring constants K 1 and K2, respectively, are arranged to
support a body A. In Fig. 1 the springs are coupled in "series" and in Fig. 2 they are in "parallel". What
are the extensions of the individual springs in these two cases as a result of the force of gravity on A?
Determine also the equivalent spring constant in the two cases.
67. A tug of war is held between two teams of five men each. Each man weighs 160 lb. and each man's pull
on the rope can be described as: F = (200 b.) e–t/τ , the mean tiring time τ is 10 sec for team A, and 20sec
for team B. If the mass of the rope is 50 lb., find the motion, that is, the final velocity of the teams. What
assumption leads to this absurd result?
68. A particle of mass m = 2kg starts from rest at the origin of an inertial coordinate system at time t = 0 . A
force F➙ = 2i˄ + 4tj˄ + 6t2 k˄ is applied to it. Find the acceleration, velocity, and position of the particle
for any later time.
69. At t = 0 , a particle of mass 1 kg has a velocity vO = 3j˄ m/sec and is at a position rO = 2k˄ m . Find the
acceleration, velocity, and position of the particle as a function of time if it is acted on by the two forces
F1 = 2i˄ + 3j˄ + 4k˄ and F2 = 1i˄ – 2j˄ – 3k˄ simultaneously.
70. A particle is projected up an inclined plane with an initial velocity v 0 of 100 cm/sec and an initial angle
θ0 of 135 degrees between the velocity and the line of maximum slope, as shown in Fig. 1. Neglecting
friction, what is the particle velocity when θ has the values 90, 45, and 0 degrees? Use path coordinates.
71. A force F➙ = ti˄ + t2j˄ + t3k˄ measured with respect to the inertial coordinate system is applied to a
particle of mass 1 kg which is initially at rest at the origin of the coordinate system. Find the accel-
eration, velocity, and displacement of the particle as functions of time, expressing your results in vector
form.
72. To a particle of mass m, initial velocity v0 , apply the force:
F(t) = 0 for t < t0
F(t) = p0/δt t0 ≤ t ≤ t0 + δt
F(t) = 0 t > t0 + δt
a. Find v(t) and x(t) .
b. Show that as δ(t) → 0 the motion approaches constant velocity with an abrupt change of velocity at
t = t0 with amount p0/m .
73. An electron is released from rest at the cathode of a vacuum tube containing two parallel plates. The
potential between cathode and anode is V volts. Determine the minimum uniform magnetic induction
field required to prevent the electron from reaching the anode. Also determine the trajectory of the
electron. Consider the electric field between cathode and anode to be uniform.
74. A particle falls toward the earth and is acted upon solely by the force of gravitation. The height from
which the particle began falling, h, is so great that the approximation F = mg cannot be used. Describe
the motion.
Motion of the Center of Mass
75. Two particles of mass, m 1 (= 3 Kg) and m2 (= 1 Kg) are attached to the ends of a rigid massless bar 40
cm. long. The system is placed, with the bar vertical and m 1 on top, on a frictionless plane. It is then
released. How far from the initial position of m2 will the mass m1 be when it hits the plane?
76. An unsymmetrical dumbbell consists of two balls, w 1 = 1 lb and w2 = 3 lb, which are connected by
a massless rod such that there is a separation of 2 ft between their centers. The dumbbell is at rest on a
frictionless table; two horizontal forces of 3 lb and 4 lb are applied at t = 0, as shown. The axis of the
dumbbell is initially along the x-axis, with the small weight at the origin. The forces F 1➙ and F2➙ remain
constant in magnitude and direction regardless of the dumbbell's motion. What are the coordinates of the
center of mass of the dumbbell after 3 sec?
Frictional Forces
77. A weight, W, is found not to slide off a rotating horizontal disk if the weight is closer than 2 ft. to the
center. The rotation rate is 21 rpm. What is the coefficient of friction between the weight and the disk?
78. As shown in the figure, MA = 2Kg , MB = 8Kg, MC = 4Kg . If the coefficient of friction between A and B
is 0.6 and zero between B and the table, show that block A will slide along the block B. Find the
accelerations of blocks A and B relative to the table.
79. A uniform straight rigid bar of mass m and length L is placed in a horizontal position across the top of
two identical cylindrical rollers, rotating as shown in Figure 1. Axes of the rollers are a distance 2d
apart. If μ is the coefficient of friction between each cylinder surface and the bar, show that if the bar is
displaced a distance x from its central position, the net horizontal force on the bar is F = –mg μx/d. Show
that the bar will execute simple harmonic motion with a period of P = 2π√(d/μg) .
80. An Eskimo is about to push along a horizontal snowfield a sled weighing 57.6 lbs carrying a baby seal
weighing 70 lbs which he has killed while hunting. The coefficient of static friction between sled and
seal is 0.8 and the coefficient of kinetic friction between sled and snow is 0.1. Show that the maximum
horizontal force that the Eskimo can apply to the sled without losing the seal is 114.8 lbs. Calculate the
acceleration of the sled when this maximum horizontal force is applied.
Note: f2➙ points to the right. The sled is moving to the right, and the seal's inertia tends to keep it at rest.
Thus any relative motion of the seal with respect to the sled would have the seal moving to the left and,
therefore, the friction would oppose this motion and point to the right.
81. A horizontal force, |F| = bt (where t is time in seconds), is applied to a block of wood of mass m at rest
on a horizontal surface. The coefficient of static friction is μs and coefficient of kinetic friction is μk .
Find the acceleration of the block of wood as a function of time.
82. The coefficient of kinetic friction is determined to be an function of velocity given in Figure 1.
Determine, as a function of time, the velocity of a 10-g mass acted upon by a horizontal force of 2.94 ×
103 dynes.
Uniform Circular Motion
83. An automobile weighing 3400 lb is driven by a man weighing 150 lb. It is moving on a circular curve in
a highway; the curve has a radius of 2000 ft. If the automobile is moving with a velocity of 60 mph, how
much centrifugal force does the man experience? Find the frictional force between the wheels and the
road.
84. The string of a conical pendulum is 10 ft long and the bob has a mass of 1/2 slug. The pendulum is
rotating at ½ rev/s. Find the angle the string makes with the vertical, and also the tension in the string.
85. The outside curve on a highway forms an arc whose radius is 150 ft. If the roadbed is 30 ft. wide and its
outer edge is 4 ft. higher than the inner edge, for what speed is it ideally banked?
86. a. A particle of mass m and charge q is injected into a uniform magnetic field, B ➙. If the particle velocity
is initially perpendicular to the field, determine its trajectory. b. Determine the cyclotron frequency of
an electron in the atmosphere.
87. A pendulum consists of a 3-ft string, fixed at one end and carrying a 4-lb body at the other. As the
pendulum swings back and forth in a vertical plane, the velocity is found to be 10 ft/sec when the
angular deflection is 45°. At this instant, what is the tension in the string?
88. A pendulum consists of a 3 ft string fixed at one end and carrying a 4 lb body at the other end. Let the
pendulum move conically, so that the body at the end of the string moves in a horizontal rather than a
vertical circle, and the string generates the surface of a cone, as shown. (Such motion is employed in
centrifugal regulators.) What speed is required to make the angle between the string and the vertical 45°,
and what is the corresponding tension in the string?
89. A string connecting a 5-lb ball and a 10-lb block is passed over an ideal pulley of negligible radius, as
shown in Figure 1. The pulley is then rotated about the axis a – a, which is assumed to pass through both
the center of the pulley and the center of gravity of the 10-lb weight. If the amount of string on the ball
side of the pulley is 3 ft. what must be the constant speed of the ball to keep the 10-lb block from
falling?
90. A sphere of weight w = 4 lbs attached to two wires BC and AC as shown (Fig 1) revolves in a horizontal
circle at a constant speed. The distance AB is 5 ft. Determine: a) the speed v for which the tension in
each wire is equal b) the value of that tension.
91. Prove that the radial acceleration experienced by a body moving in circular motion is equal to v 2/r ,
using calculus.
Central Forces and The Conservation Of Angular Momentum
92. A sphere of mass m = 3kg is attached to an elastic cord, the spring constant of the cord k = 120 n/m. At
the position P (see Fig 2) the velocity of the sphere Va➙ is perpendicular to OP, Va = 5 m/s and its
distance from the original position O (when the cord is unstretched) is a = 0.8m. Determine: a) the
maximum distance from the origin O attained by the sphere b) the corresponding speed of the sphere.
93. A small ball rolls along a horizontal circle inside a bowl at a speed V O (Fig 1). The inside surface of the
bowl is obtained by rotating the curve OA about the y axis. Assuming that the speed of the ball V O is
proportional to the distance x from the y axis to the ball determine the curve OA (Fig 2) of the inside
surface of the bowl.
94. A 2800-lb automobile moves along a highway down into a valley. The highway's path is a parabola, y =
x2/1000. What is the normal force on the car as it passes through the nadir of the curve, coordinate (0,0),
at 60 mph?
95. A particle of mass m moves according to
x = x0 + at2
y = bt3
z = ct .
Find the angular momentum L at any time t. Find the force F ➙, and from it the torque N➙, acting on the
particle. Verify that these quantities satisfy
(dL➙/dt) = r➙ × F➙ = N➙
96. A particle of mass m is repelled from the origin by a force f = k/x 3 where x is the distance from the
origin. Solve the equation of motion if the particle is initially at rest at a distance x0 from the origin.
97. Ring A of mass 8lbs is attached to a spring of constant 40 lb/ft (Fig 1). When the system is at rest the
length of the spring is 18 in and the distance between the y axis and the ring r = 12 in. The system is set
into motion with Vθ = 15 ft/sec and Vr = 0. Determine: a) the maximum distance between the origin and
the ring, b) the corresponding velocity.
