Problem 16.116* It has been suggested that a heavy
chain could be used to gradually stop an airplane that
rolls past the end of the runway. A hook attached to
the end of the chain engages the plane’s nose wheel,
and the plane drags an increasing length of the chain as
it rolls. Let mbe the airplane’s mass and v0its initial
velocity, and let ρLbe the mass per unit length of the
chain. Neglecting friction and aerodynamic drag, what
is the airplane’s velocity as a function of s?
s
2. The mass “flow” of the chain is
Use the chain rule and integrate:
ρLs
2+mdv
ds =−ρLv
2,dv
v=− ρL
2ρLs
2+mds,
ln(v) =−ln m+ρLs
2+C.
m+ρLs
2
Problem 16.117* In Problem 16.116, the frictional
force exerted on the chain by the ground would actually
dominate other forces as the distance sincreases. If the
coefficient of kinetic friction between the chain and the
ground is µkand you neglect all other forces except
the frictional force, what is the airplane’s velocity as a
function of s?
Problem 16.116. From Problem 16.116, the weight of the chain being
2. From Newton’s second law
m+ρLs
v2
2=−2µkg
ρL m+ρLs
2−mln m+ρLs
m+C.
314
Problem 16.118 A turbojet engine is being operated
on a test stand. The mass flow rate of air entering the
compressor is 13.5 kg/s, and the mass flow rate of fuel
is 0.13 kg/s. The effective velocity of air entering the
compressor is zero, and the exhaust velocity is 500 m/s.
What is the thrust of the engine? (See Example 16.12.)
dmc
dt
dmf
dt
dmc
dt
Problem 16.119 A turbojet engine is in an airplane
velocity (relative to the airplane) is 500 m/s. What is the
thrust of the engine? (See Example 16.12.)
Solution: Use the “rest frame” of the engine to determine the
The thrust is
f
Problem 16.120 A turbojet engine’s thrust reverser
causes the exhaust to exit the engine at 20◦from the
engine centerline. The mass flow rate of air entering
the compressor is 44 kg/s, and the air enters at 60 m/s.
The mass flow rate of fuel is 1.5 kg/s, and the exhaust
velocity is 370 m/s. What braking force does the engine
exert on the airplane? (See Example 16.12.)
20°
Solution:
Problem 16.121 The total external force on a 10-kg
object is constant and equal to 90i−60j+20k(N).At
t=2 s, the object’s velocity is −8i+6j(m/s).
Solution:
(a) The impulse is
Problem 16.122 The total external force on an object
is F=10ti+60j(lb). At t=0, the object’s velocity
is v=20j(ft/s).Att=12 s, the xcomponent of its
velocity is 48 ft/s.
(a) What impulse is applied to the object from t=0
to t=6s?
(b) What is the object’s velocity at t=6s?
316
Problem 16.123 An aircraft arresting system is used to
stop airplanes whose braking systems fail. The system
stops a 47.5-Mg airplane moving at 80 m/s in 9.15 s.
(a) What impulse is applied to the airplane during the
9.15 s?
(b) What is the average deceleration to which the
passengers are subjected?
Problem 16.124 The 1895 Austrain 150-mm howitzer
had a 1.94-m-long barrel, possessed a muzzle velocity
of 300 m/s, and fired a 38-kg shell. If the shell took
0.013 s to travel the length of the barrel, what average
force was exerted on the shell?
Problem 16.125 An athlete throws a shot put weighing
16 lb. When he releases it, the shot is 7 ft above the
ground and its components of velocity are vx=31 ft/s
and vy=26 ft/s.
(a) Suppose the athlete accelerates the shot from rest in
0.8 s, and assume as a first approximation that the
force Fhe exerts on the shot is constant. Use the
principle of impulse and momentum to determine
the xand ycomponents of F.
(b) What is the horizontal distance from the point
where he releases the shot to the point where it
strikes the ground?
x
y
318
Problem 16.126 The 6000-lb pickup truck Amoving
at 40 ft/s collides with the 4000-lb car Bmoving at
30 ft/s.
(a) What is the magnitude of the velocity of their
common center of mass after the impact?
(b) Treat the collision as a perfectly plastic impact.
How much kinetic energy is lost?
A
B
30°
Solution:
Problem 16.127 Two hockey players (mA=80 kg,
mB=90 kg) converging on the puck at x=0, y=
0 become entangled and fall. Before the collision,
vA=9i+4j(m/s)and vB=−3i+6j(m/s). If the
coefficient of kinetic friction between the players and
the ice is µk=0.1, what is their approximate position
when they stop sliding?
A
x
y
B
Problem 16.128 The cannon weighed 400 lb, fired a
cannonball weighing 10 lb, and had a muzzle velocity
of 200 ft/s. For the 10◦elevation angle shown, determine
(a) the velocity of the cannon after it was fired and
(b) the distance the cannonball traveled. (Neglect drag.)
10°
Solution: Assume that the height of the cannon mouth above the
g=2.16 s.
From which the range is ximpact =v0xtflight =415 ft since v0xis
constant during the flight.
Problem 16.129 A 1-kg ball moving horizontally
at 12 m/s strikes a 10-kg block. The coefficient of
restitution of the impact is e=0.6, and the coefficient
of kinetic friction between the block and the inclined
surface is µk=0.4. What distance does the block slide
before stopping? 25°
320
Problem 16.130 A Peace Corps volunteer designs the
simple device shown for drilling water wells in remote
areas. A 70-kg “hammer,” such as a section of log or
a steel drum partially filled with concrete, is hoisted to
h=1 m and allowed to drop onto a protective cap on
the section of pipe being pushed into the ground. The
combined mass of the cap and section of pipe is 20 kg.
Assume that the coefficient of restitution is nearly zero.
(a) What is the velocity of the cap and pipe
immediately after the impact?
(b) If the pipe moves 30 mm downward when the
hammer is dropped, what resistive force was
exerted on the pipe by the ground? (Assume that
the resistive force is constant during the motion of
the pipe.)
Hammer
h
Problem 16.131 A tugboat (mass =40 Mg) and a
barge (mass =160 Mg) are stationary with a slack
hawser connecting them. The tugboat accelerates to
2 knots (1 knot =1852 m/h) before the hawser becomes
taut. Determine the velocities of the tugboat and
the barge just after the hawser becomes taut (a) if
the “impact” is perfectly plastic (e =0)and (b) if the
“impact” is perfectly elastic (e =1). Neglect the forces
exerted by the water and the tugboat’s engines.
Problem 16.132 In Problem 16.131, determine the
magnitude of the impulsive force exerted on the tugboat
in the two cases if the duration of the “impact” is 4 s.
Neglect the forces exerted by the water and the tugboat’s
engines during this period.
Problem 16.133 The 10-kg mass Ais moving at 5 m/s
when it is 1 m from the stationary 10-kg mass B. The
coefficient of kinetic friction between the floor and the
two masses is µk=0.6, and the coefficient of restitution
of the impact is e=0.5. Determine how far Bmoves
from its initial position as a result of the impact.
1 m
AB
5 m /s
Solution: Use work and energy to determine A’s velocity just
322
Problem 16.134 The kinetic coefficients of friction
between the 5-kg crates Aand Band the inclined
surface are 0.1 and 0.4, respectively. The coefficient of
restitution between the crates is e=0.8. If the crates
are released from rest in the positions shown, what are
the magnitudes of their velocities immediately after they
collide?
0.1 m
60°
A
B
Problem 16.135 Solve Problem 16.134 if crate Ahas
a velocity of 0.2 m/s down the inclined surface and
crate Bis at rest when the crates are in the positions
shown.
324
Problem 16.136 A small object starts from rest at A
and slides down the smooth ramp. The coefficient of
restitution of the impact of the object with the floor is
e=0.8. At what height above the floor does the object
hit the wall?
1 ft
3 ft 60°
6 ft
A
Solution: The impact with the floor is an oblique central impact
(5) Is the ball on an upward or downward part of its path when it
Problem 16.137 The cue gives the cue ball Aa veloc-
ity of magnitude 3 m/s. The angle β=0 and the coef-
ficient of restitution of the impact of the cue ball and
the eight ball Bis e=1. If the magnitude of the eight
ball’s velocity after the impact is 0.9 m/s, what was the
coefficient of restitution of the cue ball’s impact with
the cushion? (The balls are of equal mass.)
30°
AB
β
Problem 16.138 What is the solution to Prob-
lem 16.137 if the angle β=10◦?
326
Problem 16.139 What is the solution to Prob-
lem 16.137 if the angle β=15◦and the coefficient
of restitution of the impact between the two balls is
e=0.9?
ePn =0.9659i+0.2588j.
The projection of the velocity vAP 2=vAP 3eP+vAP 3nePn. From the
solution to Problem 16.139, vAP 2=2.6i−1.5ej, from which the two
simultaneous equations for the new components: 2.6=0.2588vAP 3+
0.9659vAP 3n, and −1.5e=−0.9659vAP 3+0.2588vAP 3n.
Solve: vAP 3=0.6724 +1.449e,vAP 3n=2.51 −0.3882e. The com-
ponent of the velocity normal to the line Pis unchanged by impact.
The velocity of the 8-ball after impact is found from the conservation
of linear momentum and the coefficient of restitution:
mAvAP 3=mAv′
AP 3+mBv′
BP3,
and eB=v′
BP3−v′
AP 3
vAP 3
,
Problem 16.140 A ball is given a horizontal velocity
of 3 m/s at 2 m above the smooth floor. Determine the
distance Dbetween the ball’s first and second bounces
if the coefficient of restitution is e=0.6. 2 m
3 m /s
D
Problem 16.141* A basketball dropped from a height
of 4 ft rebounds to a height of 3 ft. In the layup shot
shown, the magnitude of the ball’s velocity is 5 ft/s, and
the angles between its velocity vector and the positive
coordinate axes are θx=42◦,θy=68◦, and θz=124◦
just before it hits the backboard. What are the magnitude
of its velocity and the angles between its velocity vector
and the positive coordinate axes just after the ball hits
the backboard?
x
y
z
v=0.866.
The ball’s velocity just before it hits the backboard is
4.81 =39.5◦,
θy=arccos 1.87
4.81 =59.8◦.
328
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2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
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Problem 16.142* In Problem 16.141, the basketball’s
diameter is 9.5 in., the coordinates of the center of basket
Solution: See the solution of Problem 16.141. The ball’s velocity
after impact in in./s is
Setting
Problem 16.143 A satellite at r0=10,000 mi from
the center of the earth is given an initial velocity
v0=20,000 ft/s in the direction shown. Determine the
magnitude of the transverse component of the satellite’s
velocity when r=20,000 mi. (The radius of the earth
is 3960 mi.)
r
45°
θ
0
r0
RE
Problem 16.144 In Problem 16.143, determine the
magnitudes of the radial and transverse components of
the satellite’s velocity when r=15,000 mi.
Solution: From the solution to Problem 16.143, the constant angu-
Problem 16.145 The snow is 2 ft deep and weighs
20 lb/ft3, the snowplow is 8 ft wide, and the truck trav-
els at 5 mi/h. What force does the snow exert on the
truck?
330
Problem 16.146 An empty 55 lb drum, 3 ft in diame-
ter, stands on a set of scales. Water begins pouring into
the drum at 1200 lb/min from 8 ft above the bottom of
the drum. The weight density of water is approximately
62.4 lb/ft3. What do the scales read 40 s after the water
starts pouring? 8 ft
62.440
60 =12.82 ft3.
The height of water in the drum is
Problem 16.147 The ski boat’s jet propulsive system
draws water in at Aand expels it at Bat 80 ft/s relative
to the boat. Assume that the water drawn in enters
with no horizontal velocity relative to the surrounding
water. The maximum mass flow rate of water through the
engine is 2.5 slugs/s. Hydrodynamic drag exerts a force
on the boat of magnitude 1.5vlb, where vis the boat’s
velocity in feet per second. Neglecting aerodynamic
drag, what is the ski boat’s maximum velocity?
AB
Problem 16.148 The ski boat of Problem 16.147
weighs 2800 lb. The mass flow rate of water through
the engine is 2.5 slugs/s, and the craft starts from rest at
t=0. Determine the boat’s velocity (a) at t=20 s and
(b) t=60 s.
Solution: Use the solution to Problem 16.147: the sum of the
forces on the boat is
Invert:
Problem 16.149* A crate of mass mslides across a
smooth floor pulling a chain from a stationary pile. The
mass per unit length of the chain is ρL. If the velocity
of the crate is v0when s=0, what is its velocity as a
function of s?
s
332