Problem 16.1 The 20-kg crate is stationary at time
t=0. It is subjected to a horizontal force given as a
function of time (in newtons) by F=10 +2t2.
(a) Determine the magnitude of the linear impulse
exerted on the crate from t=0tot=4s.
(b) Use the principle of impulse and momentum to
determine how fast the crate is moving at t=4s.
F
Solution:
(a) The impulse
(b) Use the principle of impulse and momentum
Problem 16.2 The 100-lb crate is released from rest
on the inclined surface at time t=0. The coefficient
of kinetic friction between the crate and the surface is
µk=0.18.
(a) Determine the magnitude of the linear impulse due
to the forces acting on the crate from t=0to
t=2s.
(b) Use the principle of impulse and momentum to
determine how fast the crate is moving at t=2s.
30⬚
Solution: We have
254
Problem 16.3 The mass of the helicopter is 9300 kg. It
takes off vertically at time t=0. The pilot advances the
throttle so that the upward thrust of its engine (in kN) is
given as a function of time in seconds by T=100 +2t2.
(a) Determine the magnitude of the linear impulse due
to the forces acting on the helicopter from t=0to
t=3s.
(b) Use the principle of impulse and momentum to
determine how fast the helicopter is moving at
t=3s.
Solution:
v2=4.76 m/s.
Problem 16.4 A 150 million-kg cargo ship starts from
rest. The total force exerted on it by its engines and
hydrodynamic drag (in newtons) can be approximated
as a function of time in seconds by Ft=937,500 −
0.65t2. Use the principle of impulse and momentum to
determine how fast the ship is moving in 16 minutes.
3(0.65)(960)3=7.08 ×108N-s.
Using the principle of impulse and momentum, we have
Problem 16.5 The combined mass of the motorcycle
and rider is 136 kg. The coefficient of kinetic friction
between the motorcycle’s tires and the road is µk=0.6.
The rider starts from rest and spins the rear (drive) wheel.
The normal force between the rear wheel and road is
790 N.
(a) What impulse does the friction force on the rear
wheel exert in 2 s?
(b) If you neglect other horizontal forces, what velocity
is attained by the motorcycle in 2 s?
Solution:
256
Problem 16.6 A bioengineer models the force gen-
erated by the wings of the 0.2-kg snow petrel by an
equation of the form F=F0(1+sin ωt), where F0and
ωare constants. From video measurements of a bird
taking off, he estimates that ω=18 and determines that
the bird requires 1.42 s to take off and is moving at
6.1 m/s when it does. Use the principle of impulse and
momentum to determine the constant F0.
Solution:
Problem 16.7 In Active Example 16.1, what is the
average total force acting on the helicopter from t=0
to t=10 s?
y
Solution: From Active Example 16.1 we know the total impulse
that occurs during the time. Then
Ft =I⇒F=I
t
Problem 16.8 At time t=0, the velocity of the 15-kg
object is v=2i+3j−5k(m/s). The total force acting
on it from t=0tot=4sis
F=(2t2−3t+7)i+5tj+(3t+7)k(N).
y
Solution: Working in components we have
Problem 16.9 At time t=0, the velocity of the 15-kg
object is v=2i+3j−5k(m/s). The total force acting
on it from t=0tot=4sis
F=(2t2−3t+7)i+5tj+(3t+7)k(N).
What is the average total force on the object during the
interval of time from t=0tot=4s?
y
x
兺F
0
2(4)2=40 N,
Iz=4
(3t+7)dt =3
Problem 16.10 The 1-lb collar Ais initially at rest
in the position shown on the smooth horizontal bar. At
t=0, a force
F=1
20 t2i+1
10 tj−1
30 t3k(lb)
is applied to the collar, causing it to slide along the bar.
What is the velocity of the collar at t=2s?
y
A
F
x
258
Problem 16.11 The yaxis is vertical and the curved
bar is smooth. The 4-lb slider is released from rest in
position 1 and requires 1.2 s to slide to position 2. What
is the magnitude of the average tangential force acting
on the slider as it moves from position 1 to position 2?
y
1
2 ft
Solution: We will use the principle of work and energy to find the
velocity at position 2.
Problem 16.12 During the first 5 s of the 14,200-kg
airplane’s takeoff roll, the pilot increases the engine’s
thrust at a constant rate from 22 kN to its full thrust of
112 kN.
(a) What impulse does the thrust exert on the airplane
during the 5 s?
(b) If you neglect other forces, what total time is
required for the airplane to reach its takeoff speed
of 46 m/s?
Problem 16.13 The 10-kg box starts from rest on the
smooth surface and is subjected to the horizontal force
described in the graph. Use the principle of impulse and
momentum to determine how fast the box is moving at
t=12 s.
F
50 N
F (N)
Solution: The impulse is equal to the area under the curve in the
Problem 16.14 The 10-kg box starts from rest and is
subjected to the horizontal force described in the graph.
The coefficients of friction between the box and the sur-
face are µs=µk=0.2. Determine how fast the box is
moving at t=12 s.
F
50 N
0
F (N)
t (s)
4812
Solution: The box will not move until the force Fis able to over-
260
Problem 16.15 The crate has a mass of 120 kg, and
the coefficients of friction between it and the sloping
dock are µs=0.6 and µk=0.5. The crate starts from
rest, and the winch exerts a tension T=1220 N.
(a) What impulse is applied to the crate during the first
second of motion?
(b) What is the crate’s velocity after 1 s?
30°
Solution: The motion starts only if T−mg sin 30◦>µ
smg
=121.7t=121.7 N-s
(b) The velocity is v=121.7
120 =1.01 m/s
Problem 16.16 Solve Problem 16.15 if the crate starts
from rest at t=0 and the winch exerts a tension T=
Solution: From the solution to Problem 16.15, motion will start.
Problem 16.17 In an assembly-line process, the 20-kg
package Astarts from rest and slides down the smooth
ramp. Suppose that you want to design the hydraulic
device Bto exert a constant force of magnitude Fon
the package and bring it to a stop in 0.15 s. What is the
required force F?
2 m
B
A
0.15 =590.67 N, from
which F=688.8N
N
Problem 16.18 The 20-kg package Astarts from rest
and slides down the smooth ramp. If the hydraulic
device Bexerts a force of magnitude F=540(1+
0.4t2)N on the package, where tis in seconds measured
from the time of first contact, what time is required to
bring the package to rest?
t, s
Problem 16.19 In a cathode-ray tube, an electron
(mass =9.11 ×10−31 kg)is projected at Owith
velocity v=(2.2×107)i(m/s). While the electron is
between the charged plates, the electric field generated
by the plates subjects it to a force F=−eEj. The charge
of the electron is e=1.6×10−19 C (coulombs), and the
electric field strength is E=15 sin(ωt) kN/C, where the
circular frequency ω=2×109s−1.
(a) What impulse does the electric field exert on the
electron while it is between the plates?
(b) What is the velocity of the electron as it leaves the
region between the plates?
y
Ox
++++++++
––––––––
30 mm
2.2×107=1.36 ×10−9s.
(a) The impulse is
t2
Fdt=t
(−eE) dt =t
9.11 ×10−31 =−2.52 ×106m/s.
262
Problem 16.20 The two weights are released from
rest at time t=0. The coefficient of kinetic friction
between the horizontal surface and the 5-lb weight is
µk=0.4. Use the principle of impulse and momentum
to determine the magnitude of the velocity of the 10-lb
weight at t=1s.
Strategy: Apply the principle to each weight indi-
vidually.
10 lb
5 lb
Solution:
Problem 16.21 The two crates are released from rest.
Their masses are mA=40 kg and mB=30 kg, and the
coefficient of kinetic friction between crate Aand the
inclined surface is µk=0.15. What is the magnitude of
the velocity of the crates after 1 s?
A
B
20°
Solution: The force acting to move crate Ais
Problem 16.22 The two crates are released from rest.
Their masses are mA=20 kg and mB=80 kg, and the
surfaces are smooth. The angle θ=20◦. What is the
magnitude of the velocity after 1 s?
Strategy: Apply the principle of impulse and
momentum to each crate individually.
A
B
Solution: The free body diagrams are as shown:
Subtracting the second equation from the first one,
1
0
(80 −20)(9.81)sin 20◦dt =(80 +20)v.
Solving, we get v=2.01 m/s.
at t = 1s)
Problem 16.23 The two crates are released from rest.
Their masses are mA=20 kg and mB=80 kg. The
coefficient of kinetic friction between the contacting
surfaces is µk=0.1. The angle θ=20◦. What is the
magnitude of the velocity of crate Aafter 1 s?
264
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Problem 16.24 At t=0, a 20-kg projectile is given
an initial velocity v0=20 m/s at θ0=60◦above the
horizontal.
(a) By using Newton’s second law to determine the
acceleration of the projectile, determine its velocity
at t=3s.
(b) What impulse is applied to the projectile by its
weight from t=0tot=3s?
(c) Use the principle of impulse and momentum to
determine the projectile’s velocity at t=3s.
θ
0
x
y
0
Solution:
Problem 16.25 A soccer player kicks the stationary
0.45-kg ball to a teammate. The ball reaches a maximum
height above the ground of 2 m at a horizontal distance
of 5 m form the point where it was kicked. The
duration of the kick was 0.04 seconds. Neglecting the
effect of aerodynamic drag, determine the magnitude of
the average force the player everted on the ball.
Solution: We will need to find the initial velocity of the ball. In
the ydirection we have
266
Problem 16.26 An object of mass m=2 kg slides
with constant velocity v0=4 m/s on a horizontal table
(seen from above in the figure). The object is connected
by a string of length L=1 m to the fixed point Oand
is in the position shown, with the string parallel to the
xaxis, at t=0.
(a) Determine the xand ycomponents of the force
exerted on the mass by the string as functions of
time.
(b) Use your results from part (a) and the principle of
impulse and momentum to determine the velocity
vector of the mass at t=1s.
Strategy: To do part (a), write Newton’s second law
for the mass in terms of polar coordinates.
y
x
Om
0
L
Solution:
0
Fdt =mvf−mv0
1
0
Tdt =mvxi+mvyj−m(4)j
Problem 16.27 A rail gun, which uses an electromag-
Solution:
Problem 16.28 The mass of the boat and its passenger
is 420 kg. At time t=0, the boat is moving at 14 m/s
and its motor is turned off. The magnitude of the
hydrodynamic drag force on the boat (in newtons) is
given as a function of time by 830(1 −0.08t). Determine
how long it takes for the boat’s velocity to decrease to
5 m/s.
Solution: The principle of impulse and momentum gives
Problem 16.29 The motorcycle starts from rest at t=
0 and travels along a circular track with 300-m radius.
From t=0tot=10 s, the component of the total force
on the motorcycle tangential to its path is Ft=600 N.
The combined mass of the motorcycle and rider is
150 kg. Use the principle of impulse and momentum
to determine the magnitude of the motorcycle’s velocity
at t=10 s. (See Active Example 16.2.)
Problem 16.30 Suppose that from t=0tot=10 s,
the component of the total tangential force on the
Solution:
10 s
268
Problem 16.31 The titanium rotor of a Beckman
Coulter ultracentrifuge used in biomedical research
contains 2-gram samples at a distance of 41.9 mm from
the axis of rotation. The rotor reaches its maximum
speed of 130,000 rpm in 12 minutes.
(a) Determine the average tangential force exerted
on a sample during the 12 minutes the rotor is
acceleration.
(b) When the rotor is at its maximum speed, what
normal acceleration are samples subjected to?
(12 min)60 s
min
Fave =0.00158 N
Problem 16.32 The angle θbetween the horizontal
and the airplane’s path varies from θ=0toθ=30◦
at a constant rate of 5 degrees per second. During this
maneuver, the airplane’s thrust and aerodynamic drag
are again balanced, so that the only force exerted on
the airplane in the direction tangent to its path is due
to its weight. The magnitude of the airplane’s velocity
when θ=0 is 120 m/s. Use the principle of impulse and
momentum to determine the magnitude of the velocity
when θ=30◦.
u
Solution:
Problem 16.33 In Example 16.3, suppose that the
mass of the golf ball is 0.046 kg and its diameter is
43 mm. The club is in contact with the ball for 0.0006 s,
and the distance the ball travels in one 0.001-s interval is
50 mm. What is the magnitude of the average impulsive
force exerted by the club?
Fave =mv
0.001 s
Problem 16.34 In a test of an energy-absorbing
bumper, a 2800-lb car is driven into a barrier at 5 mi/h.
The duration of the impact is 0.4 seconds. When the car
rebounds from the barrier, the magnitude of its velocity
is 1.5 mi/h.
(a) What is the magnitude of the average horizontal
force exerted on the car during the impact?
(b) What is the average deceleration of the car during
the impact?
270
Problem 16.35 A bioengineer, using an instrumented
dummy to test a protective mask for a hockey goalie,
launches the 170-g puck so that it strikes the mask
moving horizontally at 40 m/s. From photographs of
the impact, she estimates the duration to be 0.02 s and
observes that the puck rebounds at 5 m/s.
(a) What linear impulse does the puck exert?
(b) What is the average value of the impulsive force
exerted on the mask by the puck?
Problem 16.36 A fragile object dropped onto a hard
surface breaks because it is subjected to a large impul-
sive force. If you drop a 2-oz watch from 4 ft above
the floor, the duration of the impact is 0.001 s, and the
watch bounces 2 in. above the floor, what is the average
value of the impulsive force?
Problem 16.37 The 0.45-kg soccer ball is given a kick
with a 0.12-s duration that accelerates it from rest to a
velocity of 12 m/s at 60◦above the horizontal.
(a) What is the magnitude of the average total force
exerted on the ball during the kick?
(b) What is the magnitude of the average force exerted
on the ball by the player’s foot during the kick?
FAV (0.12)=0.45(12 cos 60◦i+12 sin 60◦j)
FAV =22.5i+39.0jN
(a) |FAV |=45.0N mg =(0.45)(9.81)=4.41 N
FAV =FFOOT +FG
FAV =FFOOT −mgj
FFOOT =FAV +mgj
FFOOT =22.5i+39.0j+4.41j
(b) |FFOOT|=48.9N
mg
N
Problem 16.38 An entomologist measures the motion
of a 3-g locust during its jump and determines that the
insect accelerates from rest to 3.4 m/s in 25 ms (mil-
liseconds). The angle of takeoff is 55◦above the hori-
zontal. What are the horizontal and vertical components
of the average impulsive force exerted by the locust’s
hind legs during the jump?
272
Problem 16.39 A 5-oz baseball is 3 ft above the
ground when it is struck by a bat. The horizontal distance
to the point where the ball strikes the ground is 180 ft.
Photographs studies indicate that the ball was moving
approximately horizontally at 100 ft/s before it was
struck, the duration of the impact was 0.015 s, and the
ball was traveling at 30◦above the horizontal after it was
struck. What was the magnitude of the average impulsive
force exerted on the ball by the bat?
30°
x=(v2cos 30◦)t
where v2is the magnitude of the velocity at the point of leaving the
bat, and y0=3 ft. At x=180 ft, t=180/(v2cos 30◦), and y=0.
Substitute and reduce to obtain
v2=180g
2 cos230◦(180 tan 30◦+y0)=80.62 ft/s.
26.42j, from which
Problem 16.40 Paraphrasing the official rules of rac-
quetball, a standard racquetball is 2 1
4inches in diameter,
weighs 1.4 ounces (16 ounces =1 pound), and bounces
between 68 and 72 inches from a 100-inch drop at
a temperature between 70 and 74 degrees Fahrenheit.
Suppose that a ball bounces 71 inches when it is dropped
from a 100-inch height. If the duration of the impact is
0.08 s, what average force is exerted on the ball by the
floor?