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Problem 15.118 The driver of a 3000-lb car moving at
40 mi/h applies an increasing force on the brake pedal.
The magnitude of the resulting frictional force exerted
on the car by the road is f=250 +6slb, where sis the
car’s horizontal position (in feet) relative to its position
when the brakes were applied. Assuming that the car’s
tires do not slip, determine the distance required for the
car to stop
(a) by using Newton’s second law and
(b) by using the principle of work and energy.
Solution:
3=41.67,c =−Wv2
6g=−53493.6.
(b) Principle of work and energy: The energy of the car when the
b=125
3=41.67,c =−Wv2
1
6g=−53493.6.
Problem 15.119 Suppose that the car in Prob-
lem 15.118 is on wet pavement and the coefficients of
friction between the tires and the road are µs=0.4 and
Problem 15.120 An astronaut in a small rocket vehicle
(combined mass =450 kg) is hovering 100 m above the
surface of the moon when he discovers that he is nearly
out of fuel and can exert the thrust necessary to cause
the vehicle to hover for only 5 more seconds. He quickly
considers two strategies for getting to the surface:
(a) Fall 20 m, turn on the thrust for 5 s, and then fall
the rest of the way;
(b) fall 40 m, turn on the thrust for 5 s, and then fall
the rest of the way.
Which strategy gives him the best chance of surviving?
How much work is done by the engine’s thrust in each
case? (gmoon =1.62 m/s2)
where F=mg, acting upward, h3is the altitude at the end of
the thrusting phase. The energy condition at the end of the thrust-
m=11.8 m/s .
He should choose strategy (b) since the impact velocity is reduced
238
Problem 15.121 The coefficients of friction between
the 20-kg crate and the inclined surface are µs=0.24
and µk=0.22. If the crate starts from rest and the hor-
izontal force F=200 N, what is the magnitude of the
velocity of the crate when it has moved 2 m?
F
(F cos 30◦−mg sin 30◦−µkN)(2m) =1
2mv2
2−0,
we obtain v2=1.77 m/s.
x
Problem 15.122 The coefficients of friction between
izontal force F=40 N. What is the magnitude of the
velocity of the create when it has moved 2 m?
Solution: See the solution of Problem 15.121. The normal force is
The friction force necessary for equilibrium is
Problem 15.123 The Union Pacific Big Boy locomo-
tive weighs 1.19 million lb, and the traction force (tan-
gential force) of its drive wheels is 135,000 lb. If you
neglect other tangential forces, what distance is required
for the train to accelerate from zero to 60 mi/h?
Problem 15.124 In Problem 15.123, suppose that the
acceleration of the locomotive as it accelerates from
zero to 60 mi/h is (F0/m)(1−v/88), where F0=
135,000 lb, mis the mass of the locomotive, and vis
its velocity in feet per second.
(a) How much work is done in accelerating the train
to 60 mi/h?
(b) Determine the locomotive’s velocity as a function
of time.
Solution: [Note: Fis not a force, but an acceleration, with the
Check: To demonstrate that this is a correct expression, it is used
240
Problem 15.125 A car traveling 65 mi/h hits the crash
barrier described in Problem 15.14. Determine the max-
imum deceleration to which the passengers are subjected
if the car weighs (a) 2500 lb and (b) 5000 lb.
s
Solution: From Problem 15.14 we know that the force in the crash
2mv2+s
0
1
(120s+40s3)ds=0
s=13.6ft,a =F
32.2 ft/s2and solving, we find that
s=16.2ft,a =F
m=120s+40s3
m=1110 ft/s2
(a)1320 ft/s2,(b)1110 ft/s2.
0
0
0.1
0.2
0.05 0.1
0.15
Problem 15.126 In a preliminary design for a mail-
sorting machine, parcels moving at 2 ft/s slide down a
smooth ramp and are brought to rest by a linear spring.
What should the spring constant be if you don’t want the
10-lb parcel to be subjected to a maximum deceleration
greater than 10g’s?
2 ft/s
3 ft
k
242
Problem 15.127 When the 1-kg collar is in position 1,
the tension in the spring is 50 N, and the unstretched
length of the spring is 260 mm. If the collar is pulled
to position 2 and released from rest, what is its velocity
when it returns to position 1? 300 mm
600 mm
12
v1=k
m(S2
2−S2
1)=14.46 m/s
600 mm
Problem 15.128 When the 1-kg collar is in position 1,
the tension in the spring is 100 N, and when the collar
is in position 2, the tension in the spring is 400 N.
(a) What is the spring constant k?
(b) If the collar is given a velocity of 15 m/s at position
1, what is the magnitude of its velocity just before
it reaches position 2?
k,0.3−S0=100
k.
from which
Problem 15.129 The 30-lb weight is released from
rest with the two springs (kA=30 lb/ft, kB=15 lb/ft)
unstretched.
(a) How far does the weight fall before rebounding?
(b) What maximum velocity does it attain?
kA
kB
244
Problem 15.130 The piston and the load it supports
are accelerated upward by the gas in the cylinder. The
total weight of the piston and load is 1000 lb. The cylin-
der wall exerts a constant 50-lb frictional force on the
piston as it rises. The net force exerted on the piston
by pressure is (p2−patm)A, where pis the pressure of
the gas, patm =2117 lb/ft2is the atmospheric pressure,
and A=1ft
2is the cross-sectional area of the piston.
Assume that the product of pand the volume of the
cylinder is constant. When s=1 ft, the piston is sta-
tionary and p=5000 lb/ft2. What is the velocity of the
piston when s=2 ft?
s
Piston
Gas
The potential energy due to gravity is
Vgravity =−
s
s0
(−W)ds =W(s −s0).
The work done by the friction is
Ufriction =s
s0
(−f)ds =−f(s−s0), where f=50 lb.
from which v=2(298.7)g
W=4.39 ft/s
Problem 15.131 When a 22,000-kg rocket’s engine
burns out at an altitude of 2 km, the velocity of the
rocket is 3 km/s and it is traveling at an angle of 60◦
relative to the horizontal. Neglect the variation in the
gravitational force with altitude.
(a) If you neglect aerodynamic forces, what is the
magnitude of the velocity of the rocket when it
reaches an altitude of 6 km?
(b) If the actual velocity of the rocket when it reaches
an altitude of 6 km is 2.8 km/s, how much work is
done by aerodynamic forces as the rocket moves
from 2 km to 6 km altitude?
Problem 15.132 The 12-kg collar Ais at rest in the
position shown at t=0 and is subjected to the tangen-
tial force F=24 −12t2N for 1.5 s. Neglecting friction,
what maximum height hdoes the collar reach?
h
A
F
m[12t2−t4]1.5
Problem 15.133 Suppose that, in designing a loop for
a roller coaster’s track, you establish as a safety criterion
that at the top of the loop the normal force exerted on a
passenger by the roller coaster should equal 10 percent of
the passenger weight. (That is, the passenger’s “effective
weight” pressing him down into his seat is 10 percent
of his actual weight.) The roller coaster is moving at
62 ft/s when it enters the loop. What is the necessary
instantaneous radius of curvature ρof the track at the
top of the loop?
ρ
50 ft
246
Problem 15.134 A 180-lb student runs at 15 ft/s, grabs
a rope, and swings out over a lake. He releases the rope
when his velocity is zero.
(a) What is the angle θwhen he releases the rope?
(b) What is the tension in the rope just before he
release it?
(c) What is the maximum tension in the rope?
θ
30 ft
Problem 15.135 If the student in Problem 15.134
releases the rope when θ=25◦, what maximum height
does he reach relative to his position when he grabs the
rope?
Problem 15.136 A boy takes a running start and jumps
on his sled at position 1. He leaves the ground at
position 2 and lands in deep snow at a distance of
b=25 ft. How fast was he going at 1?
1
2
15 ft 35°
v1=4.72 ft/s The other values were v2=25.8 ft/s, and the time in the
air before impact was timpact =1.182 s. Check: An analytical solution
is found as follows: Combine (1) and (2)
2sin θcos θ
Problem 15.137 In Problem 15.136, if the boy starts
at 1 going 15 ft/s, what distance bdoes he travel through
the air?
248
Problem 15.138 The 1-kg collar Ais attached to the
linear spring (k =500 N/m)by a string. The collar starts
from rest in the position shown, and the initial tension in
the spring is 100 N. What distance does the collar slide
up the smooth bar?
A
k
k=0.2m.
Problem 15.139 The masses mA=40 kg and mB=
60 kg. The collar Aslides on the smooth horizontal bar.
The system is released from rest. Use conservation of
energy to determine the velocity of the collar Awhen it
has moved 0.5 m to the right.
A
B
Problem 15.140 The spring constant is k=850 N/m,
mA=40 kg, and mB=60 kg. The collar Aslides on
the smooth horizontal bar. The system is released from
rest in the position shown with the spring unstretched.
Use conservation of energy to determine the velocity of
the collar Awhen it has moved 0.5 m to the right.
A
k
0.4 m
B
moved 0.5 m. The component of A′svelocity parallel to the cable
equals B′svelocity: vAcos 45◦=vB.B
′sdownward displacement
Problem 15.141 The yaxis is vertical and the curved
bar is smooth. If the magnitude of the velocity of the
4-lb slider is 6 ft/s at position 1, what is the magnitude
of its velocity when it reaches position 2?
y
x
1
2
2 ft
4 ft
Solution: Choose the datum at position 2. At position 2, the
250
Problem 15.142 In Problem 15.141, determine the
magnitude of the velocity of the slider when it reaches
position 2 if it is subjected to the additional force F=
3xi−2j(lb) during its motion.
Solution:
Problem 15.143 Suppose that an object of mass mis
beneath the surface of the earth. In terms of a polar
coordinate system with its origin at the earth’s center, the
gravitational force on the object is −(mgr/RE)er, where
REis the radius of the earth. Show that the potential
energy associated with the gravitational force is V=
mgr 2/2RE.
Solution: By definition, the potential associated with a force Fis
F·dr.
Problem 15.144 It has been pointed out that if tunnels
could be drilled straight through the earth between points
on the surface, trains could travel between these points
using gravitational force for acceleration and decelera-
tion. (The effects of friction and aerodynamic drag could
be minimized by evacuating the tunnels and using mag-
netically levitated trains.) Suppose that such a train travels
from the North Pole to a point on the equator. Determine
the magnitude of the velocity of the train
(a) when it arrives at the equator and
(b) when it is halfway from the North Pole to the equa-
tor. The radius of the earth is RE=3960 mi.
N
(See Problem 15.143.)
Solution: The potential associated with gravity is
Problem 15.145 In Problem 15.123, what is the max-
imum power transferred to the locomotive during its
acceleration?
252
Problem 15.146 Just before it lifts off, the 10,500-kg
airplane is traveling at 60 m/s. The total horizontal force
exerted by the plane’s engines is 189 kN, and the plane
is accelerating at 15 m/s2.
(a) How much power is being transferred to the plane
by its engines?
(b) What is the total power being transferred to the
plane?
Problem 15.147 The “Paris Gun” used by Germany in
World War I had a range of 120 km, a 37.5-m barrel, a
muzzle velocity of 1550 m/s and fired a 120-kg shell.
(a) If you assume the shell’s acceleration to be
constant, what maximum power was transferred to
the shell as it traveled along the barrel?
(b) What average power was transferred to the shell?
v=F
mt+C.
just before the muzzle exit: P=F(1550)=5.96 ×109joule/s =
5.96 GW. (b) From Eq. (15.18) the average power transfer is
1
2mv2
2−1
2mv2
1
