PROBLEM 14.104
In a rocket, the kinetic energy imparted to the consumed and ejected fuel is wasted as far as propelling the
rocket is concerned. The useful power is equal to the product of the force available to propel the rocket and
the speed of the rocket. If v is the speed of the rocket and u is the relative speed of the expelled fuel, show that
the mechanical efficiency of the rocket is 22
2/( ).uv u v
Explain why 1
when .uv
PROBLEM 14.105
Three identical cars are being unloaded from an
automobile carrier. Cars B and C have just been
unloaded and are at rest with their brakes off when car A
leaves the unloading ramp with a velocity of 5.76 ft/s
and hits car B, which hits car C. Car A then again hits
car B. Knowing that the velocity of car B is 5.04 ft/s
after the first collision, 0.630 ft/s after the second
collision, and 0.709 ft/s after the third collision,
determine (a) the final velocities of cars A and C, (b) the
coefficient of restitution for each of the collisions.
33
( ) ( ) 0.720 0.630
AB
PROBLEM 14.106
A 30-g bullet is fired with a velocity of 480 m/s into block A,
which has a mass of 5 kg. The coefficient of kinetic friction
between block A and cart BC is 0.50. Knowing that the cart
has a mass of 4 kg and can roll freely, determine (a) the final
velocity of the cart and block, (b) the final position of the
block on the cart.
f
PROBLEM 14.107
An 80-Mg railroad engine A coasting at 6.5 km/h strikes a 20-Mg flatcar C carrying a 30-Mg load B that can
slide along the floor of the car
(0.25).
k
Knowing that the car was at rest with its brakes released and that
it automatically coupled with the engine upon impact, determine the velocity of the car (a) immediately after
impact, (b) after the load has slid to a stop relative to the car.
SOLUTION
The masses are the engine
3
(8010kg),
A
m the load
3
(3010kg),
B
m and the flat car
3
(2010kg).
C
m
Af Bf Cf A B C f
PROBLEM 14.107 (Continued)
Equating (1) and (3) and solving for ,
f
v
0
3
3
()
(80 10 )(1.80556)
(130 10 )
AA
f
ABC
mv
vmmm
PROBLEM 14.108
In a game of pool, ball A is moving with a velocity
v
0
when it strikes
balls B and C which are at rest and aligned as shown. Knowing that
after the collision the three balls move in the directions indicated and
that
0
12 ft/sv
and
6.29 ft/s,
C
v
determine the magnitude of the
velocity of (a) ball A, (b) ball B.
SOLUTION
PROBLEM 14.109
Mass C, which has a mass of 4 kg, is suspended from a cord attached to cart
A, which has a mass of 5 kg and can roll freely on a frictionless horizontal
track. A 60-g bullet is fired with a speed v0 500 m/s and gets lodged in
block C. Determine (a) the velocity of C as it reaches its maximum elevation,
(b) the maximum vertical distance h through which C will rise.
PROBLEM 14.109 (Continued)
2233
mm
vImp v
3
22
(4.06)(9.81) 39.829 h
97.871 0 43.857 39.829 h
BC
Vmgh h
1.356 mh
Another method: We observe that no external horizontal forces are exerted on the system consisting of A, B,
and C. Thus the horizontal component of the velocity of the mass center remains constant.
5 0.06 4 9.06 kg
ABC
mmmm
PROBLEM 14.109 (Continued)
Immediately after the impact of B on C, the velocity vA is zero.
()( )
BC ABCx
mmv mmmv
PROBLEM 14.110
A 15-lb block B is at rest and a spring of constant
k
72
lb/in. is held compressed 3 in. by a cord. After
5-lb block A is placed against the end of the spring, the
cord is cut causing A and B to move. Neglecting
friction, determine the velocities of blocks A and B
immediately after A leaves B.
SOLUTION
2
2
50.15528 lb s /ft
32.2
15 0.46584 lb s /ft
32.2
A
B
m
m
2
22
2
2
22
(5)(0.5) 2.5 ft lb
11
22
11
(0.15528) (0.46584) 0.10352
223
gA
AA BB
A
AA
VWh
Tmv mv
v
vv
PROBLEM 14.110 (Continued)
2
B
PROBLEM 14.111
Car A of mass 1800 kg and car B of mass 1700 kg
are at rest on a 20-Mg flatcar which is also at rest.
Cars A and B then accelerate and quickly reach
constant speeds relative to the flatcar of 2.35 m/s
and 1.175 m/s, respectively, before decelerating to
a stop at the opposite end of the flatcar. Neglecting
friction and rolling resistance, determine the
velocity of the flatcar when the cars are moving at
constant speeds.
PROBLEM 14.112
The nozzle shown discharges water at the rate of 200 gal/min. Knowing
that at both B and C the stream of water moves with a velocity of
magnitude 100 ft/s, and neglecting the weight of the vane, determine the
force-couple system that must be applied at A to hold the vane in place
3
(1 ft 7.48 gal).
SOLUTION
3
3
33
200 gal/min
(7.48 gal/ft )(60 s/min)
0.44563 ft /s
(62.4 lb/ft )(0.44563 ft /s)
Q
dm Q
dt g
PROBLEM 14.112 (Continued)
PROBLEM 14.113
An airplane with weight W and total wing
span b flies horizontally at a constant speed v.
Use the airplane as a reference frame; that is,
consider the airplane to be motionless and the
air to flow past it with speed v. Suppose that a
cylinder of air with diameter b is deflected
downward by the wing (the cross section of
the cylinder is the dashed circle in in the
figure). Show that the angle through which the
cylinder stream is deflected (called the
downwash angle) is determined by the
formula 4/ where
is the
mass density of the air.
4
bv
PROBLEM 14.114
The final component of a conveyor system
receives sand at a rate of 100 kg/s at A and
discharges it at B. The sand is moving horizontally
at A and B with a velocity of magnitude
4.5
AB
vv m/s. Knowing that the combined
weight of the component and of the sand it
supports is 4W
kN, determine the reactions at
C and D.
0.9 3 1.8 1.65
mv D t W t mv
PROBLEM 14.115
A garden sprinkler has four rotating arms, each of which
consists of two horizontal straight sections of pipe forming
an angle of 120° with each other. Each arm discharges
water at a rate of 20 L/min with a velocity of 18 m/s
relative to the arm. Knowing that the friction between the
moving and stationary parts of the sprinkler is equivalent to
a couple of magnitude
0.375 N m,M
determine the
constant rate at which the sprinkler rotates.
SOLUTION
The flow through each arm is 20 L/min.
63
3
36
20 L/min 1min 333.33 10 m /s
60 s
1000 L/m
0.33333 kg/s
Q
dm Q
dt
Consider the moment about O exerted on the fluid stream of one arm. Apply the impulse-momentum
principle. Compute moments about O. First, consider the geometry of triangle OAB. Using first the law of
100 217.95
23.413 , 60 36.587
Moments about O:
PROBLEM 14.116
A chain of length l and mass m falls through a small hole in a
plate. Initially, when y is very small, the chain is at rest. In
each case shown, determine (a) the acceleration of the first
link A as a function of y, (b) the velocity of the chain as the
last link passes through the hole. In case 1, assume that the
individual links are at rest until they fall through the hole; in
case 2, assume that at any instant all links have the same
speed. Ignore the effect of friction.
SOLUTION
Let
be the mass per unit length of chain. Assume that the weight of any chain above the hole is supported by
the floor. It and the corresponding upward reaction of the floor are not shown in the diagrams.
Case 1: Apply the impulse-momentum principle to the entire chain.
()()
yv gy t y y v v
PROBLEM 14.116 (Continued)
dv