CHAPTER
8
Systems of Particles and Conservation of Momentum
1* · Give an example of a three-dimensional object that has no mass at its center of mass.
A hollow sphere.
2
· Three point masses of 2 kg each are located on the x axis at the origin, x = 0.20 m, and x = 0.50 m. Find the center
of mass of the system.
Use Equ. 8-4; note that ycm = 0
xcm = [(2 × 0 + 2 × 0.2 + 2 × 0.5)/6] m = 0.233 m
3
· A 24-kg child is 20 m from an 86-kg adult. Where is the center of mass of this system?
Take the origin at the position of the child.
Use Equ. 8-4
xcm = (86 × 20/110) m = 15.6 m
4
· Three objects of 2 kg each are located in the xy plane at points (10 cm, 0), (0, 10 cm), and (10 cm, 10 cm). Find
the location of the center of mass.
Use Equ. 8-4
xcm = [(10 × 2 + 10 × 2)/6] cm = 6.67 cm;
ycm = [(10 × 2 + 10 × 2)/6] cm = 6.67 cm
5* · Find the center of mass xcm of the three masses in Figure 8-46.
Use Equ. 8-4
xcm = [(1 × 1 + 2 × 2 + 8 × 4)/11] m = 3.36 m
6
· Alley Oop’s club-ax consists of a symmetrical 8-kg stone attached to the end of a uniform 2.5-kg stick that is 98
cm long. The dimensions of the club-ax are shown in Figure 8-47. How far is the center of mass from the handle end
of the club-ax?
1. Locate CM of stic k and of stone
By symmetry, xcm(stick) = 0.49 m; xcm(stone) = 0.89 m
2. Use Equ. 8-4
xcm = [(2.5 × 0.49 + 8 × 0.89)/10.5] m = 0.795 m
7
· Three balls A, B, and C, with masses of 3 kg, 1 kg, and 1 kg, respectively, are connected by massless rods. The
balls are located as in Figure 8-48. What are the coordinates of the center of mass?
Use Equ. 8-4
xcm = [(3 × 2 + 1 × 1 + 1 × 3)/5] m = 2 m
Chapter 8
Systems of Particles and Conservation of Momentum
ycm = [(3 × 2 + 1 × 1 + 1 × 0)/5] m = 1.4 m
8
·
By symmetry, locate the center of mass of an equilateral triangle of side length a with one vertex on the y axis
and the others at (-a/2, 0) and (+a/2, 0).
1. Draw the triangle; assume vertex at y > 0
2. Locate the intersection of the bisectors; see sketch
xcm = 0; ycm = 1/2a tan 30o = 0.289a
3. Give the coordinates of CM
9* ·· The uniform sheet of plywood in Figure 8-49 has a mass of 20 kg. Find its center of mass. We shall consider this
as two sheets, a square sheet of 3 m side length and mass m1 and a rectangular sheet 1m × 2m with a mass of -m2.
Let coordinate origin be at lower left hand corner of the sheet. Let σ be the surface density of the sheet.
1. Find xcm(m1), ycm(m1) and xcm(m2), ycm(m2)
By symmetry, xcm(m1) = 1.5 m, ycm(m1) = 1.5 m
and xcm(m2) = 1.5 m, ycm(m2) = 2.0 m
2. Determine m1 and m2
m1 = 9σ kg, m2 = 2σ kg
3. Use Equ. 8-4
xcm = (9σ × 1.5 - 2σ × 1.5)/7σ = 1.5 m
ycm = (9σ × 1.5 - 2σ × 2.0)/7σ = 1.36 m
10 ·· Show that the center of mass of a uniform semicircular disk of radius R is at a point (4/3π )R from the center of the
circle.
1. The semicircular disk is shown; we also show here the surface element dA
2. Use Equ. 8-5 to find ycm; xcm = 0 by symmetry. ycm = (1/M)∫yσ dA
3. y = r sin θ, dA = r dθdr, and M = π R2σ/2; make the appropriate
substitutions
σ R π 2
2σ R 2
2σ 3
4
θ d θ dr =
4. ycm =
r
sin
r dr =
R =
R0
0
0
0
M
M
3M
3π
∫ ∫
∫
11 ·· A baseball bat of length L has a peculiar linear density (mass per unit length) given by λ = λ0(1 + x2/L2). Find the x
coordinate of the center of mass in terms of L.
Use Equ. 8-5. xcm = ∫x dm/∫dm. Here M = ∫dm = ∫λ dx =
and
∫
L
0
xdm =
∫
L
0
∫
L
0
λ 0 (1+x2/L2)dx = 4 λ0 L/30
xλ0 (1 + x2/L2)dx = 3 λ0 L2/4.0 We find that xcm = (9/16)L.
Chapter 8
Systems of Particles and Conservation of Momentum
12 ··· Find the center of mass of a homogeneous solid hemisphere of radius R and mass M.
The volume element for a sphere is dV = r2 sin θ d θ d φ dr, where θ is the polar angle and φ the azimuthal angle.
Let the base of the hemisphere be the xy plane and ρ be the mass density. Then z = r cos θ. Now use Equ. 8-5:
ρ
zcm =
M
R π / 2 2π
∫ ∫ ∫
0
0
r3 sin θ cos θ d θ d Φ dr =
0
πρR 4  1 2 
sin θ 
2 M  2

π /2
=
0
14
πρR 4
3
3
; M =  πρR  ; zcm = R.
23
4M
8

13* ··· Find the center of mass of a thin hemispherical shell.
The element of area on the shell is dA = 2π R2 sin θ d θ, where R is the radius of the hemisphere. Let σ be the
surface mass density. Then M = 1/2(4π R2σ) = 2π R2σ. Use the same coordinates as in Problem 8-12, and apply
Equ. 8-5:
zcm =
∫ zσdA =
M
2πR 3σ
2πR 2σ
π /2
∫
sin θ cos θ d θ = R/2.
0
14 ··· A sheet of metal is cut in the shape of a parabola. The edge of the sheet is given by the expression y = ax2, and y
ranges from y = 0 to y = b. Find the center of mass in terms of
a and b.
The parabolic sheet is shown here. Note that by
symmetry, xcm = 0.
The element of area is dA = xdy, and x = y1/2/ a 0.
2 5/2
b
3
0
0
5
ycm = b
= b
= 2
= b.0
1/ 2
5
xdy
y dy
b3 / 2
0
0
3
b
∫
∫
xydy
b
∫
∫
y 3/ 2 dy
15 · On the night before your physics exam, you hear a banging on your door, and in walks Kelly. She says, “There’s a
big problem here. According to Newtonian physics, only external forces can cause the center of mass of a system to
accelerate. But a car accelerates because of its own engine, so obviously Newton was wrong.” She crosses her arms
in a way that suggests that she is not going anywhere until she gets a satisfactory explanation. How can you explain
Kelly’s error to her in order to rescue Newton and get back to your studying?
The external force here is the force of static friction between the tires and the road.
16 · Two pucks of mass m1 and m2 lie unconnected on a frictionless table. A horizontal force F1 is exerted on m1 only.
What is the magnitude of the acceleration of the center of mass of the pucks? (a) F1/m1 (b) F1/(m1 + m2) (c) F1/m2
(d) (m1 + m2)F1/m1m2
Chapter 8
Systems of Particles and Conservation of Momentum
(b) by application of Equ. 8-10.
17* · The two pucks in Problem 16 are lying on a frictionless table and connected by a spring of force constant k. A
horizontal force F1 is again exerted only on m1 along the spring away from m2.What is the magnitude of the
acceleration of the center of mass? (a) F1/m1 (b) F1/(m1 + m2) (c) (F1 + k ∆x)/(m1 + m2), where ∆x is the amount the
spring is stretched. (d) (m1 + m2)F1/m1m2
(b) by application of Equ. 8-10; the spring force is an internal force.
18 · Two 3-kg masses have velocities v1 = 2 m/s i + 3 m/s j and v2 = 4 m/s i - 6 m/s j. Find the velocity of the center of
mass for the system.
Use Equ. 8-13; vcm = (Σ mivi)/M = (v1 + v2)/2.
vcm = 3 m/s i - 1.5 m/s j
19 · A 1500-kg car is moving westward with a speed of 20 m/s, and a 3000-kg truck is traveling east with a speed of
16 m/s. Find the velocity of the center of mass of the system.
Let east be the positive i direction. vcm = (Σ mivi)/M
vcm = [(3000 × 16 - 1500 × 20) /4500] m/s i = 4 m/s i
20 · A force F = 12 N i is applied to the 3-kg ball in Problem 7. What is the acceleration of the center of mass?
2
2
Use Equ. 8-10
a cm = (12/5) m/s i = 2.4 m/s i
21* ·· A block of mass m is attached to a string and suspended inside a hollow box of mass M. The box rests on a scale
that measures the system’s weight. (a) If the string breaks, does the reading on the scale change? Explain your
reasoning. (b) Assume that the string breaks and the mass m falls with constant acceleration g. Find the acceleration
of the center of mass, giving both direction and magnitude. (c) Using the result from (b), determine the reading on the
scale while m is in free fall.
(a) Yes; initially the scale reads (M + m)g; while m is in free fall, the reading is Mg.
(b) a cm = mg/(M + m), directed downward.
(c) Fnet = (M + m)g - (M + m)a cm = Mg.
22 ·· A vertical spring of force constant k is attached at the bottom to a platform of mass mp, and at the top to a
massless cup, as in Figure 8-50. The platform rests on a scale. A ball of mass mb is placed in the cup. What is the
reading on the scale when (a) the spring is compressed an amount d = mbg/k ? (b) the ball comes to rest momentarily
with the spring compressed? (c) the ball again comes to rest in its original position?
(a) The force exerted by the spring on mp = mbg
Scale reading = mpg + mbg
2
(b) When the ball is at rest, mbgd = 1/2kd ;
Scale reading = kd + mpg = (2mb + mp)g
d = 2mbg/k
(c) In this case, the spring exerts no force on mb
Scale reading = mpg
23 ·· In the Atwood’s machine in Figure 8-51, the string passes over a fixed, frictionless cylinder of mass mc. (a) Find
the acceleration of the center of mass of the two-block-and-cylinder system. (b) Use Newton’s second law for
systems to find the force F exerted by the support. (c) Find the tension in the string connecting the blocks and show
that F = mcg + 2T.
(a) 1. Use Equ. 8-8 and results of Problem 4-81
a cm = (m1a - m2a)/(m1 + m2 + mc);
Chapter 8
Systems of Particles and Conservation of Momentum
a = (m1 - m2)/(m1 + m2)
a cm = [(m1 - m2)2g]/[(m1 + m2)(m1 + m2 +mc)]
F = Mg - (m1 - m2)2g/(m1 + m2)
= [4m1m2/(m1 + m2) + mc]g
F = 2T + mc g = [4m1m2/(m1 + m2) + mc]g; Q.E.D.
2. Simplify the expression for a cm
(b) F = Mg - Ma cm; M = m1 + m2 + mc
(c) From Problem 4-81, T = 2m1m2g/(m1 + m2)
24 ·· Repeat Problems 22a and 22b with the ball dropped into the cup from a height h above the cup.
(a) There is no change here.
Scale reading = (mp + mb)g
(b) See Prob. 7-96; xmax =
mb g 
2kh
1+ 1+

k 
mb g

0



F = mpg + mbg 1 + 1 +

2 kh
mb g

0


25* · True or false: (a) The momentum of a heavy object is greater than that of a light object moving at the same speed.
(b) The momentum of a system may be conserved even when mechanical energy is not. (c) The velocity of the center
of mass of a system equals the total momentum of the system divided by its total mass.
(a) True (for magnitude)
(b) True (inelastic collision)
(c) True
26 ·
How is the recoil of a rifle related to momentum conservation?
p recoil + p gun = 0; p recoil = -p gun.
27 · A man is stranded in the middle of an ice rink that is perfectly frictionless. How can he get to the edge?
If he throws something forward, he will move backward.
28 · A girl jumps from a boat to a dock. Why does she have to jump with more energy than she would need if she were
jumping the same distance from one dock to another?
She must give the boat a recoil momentum, and Eboat = p boat 20/2mboat .
29* ·· Much early research in rocket motion was done by Robert Goddard, physics professor at Clark College in
Worcester, Mass. A quotation from a 1921 editorial in the New York Times illustrates the public acceptance of his
work: “That Professor Goddard with his ‘chair’ at Clark College and the countenance of the Smithonian Institution
does not know the relation between action and reaction, and the need to have something better than a vacuum against
which to react—to say that would be absurd. Of course, he only seems to lack the knowledge ladled out daily in high
schools.” The belief that a rocket needs something to push against was a prevalent misconception before rockets in
space were commonplace. Explain why that belief is wrong.
Conservation of momentum does not require the presence of a medium such as air.
30 ·· Liz, Jay, and Tara discover that sinister chemicals are leaking at a steady rate from a hole in the bottom of a
railway car. To collect evidence of a potential environmental mishap, they videotape the car as it rolls without friction
at an initial speed v0. Tara claims that careful analysis of the videotape will show that the car’s speed is increasing,
because it is losing mass as it drains. The increase in speed will help to prove that the leak is occurring. Liz says no,
that with a loss of mass, the car’s speed will be decreasing. Jay says the speed will remain the same. (a) Who is right?
(b) What forces are exerted on the system of the car plus chemical cargo?
Chapter 8
Systems of Particles and Conservation of Momentum
(a) Jay is right.
(b) No net external force acts on the car plus cargo.
31 · A girl of mass 55 kg jumps off the bow of a 75-kg canoe that is initially at rest. If her velocity is 2.5 m/s to the
right, what is the velocity of the canoe after she jumps?
Fext = 0; Σ mivi = 0
vc = (55 × 2.5/75) m/s = 1.83 m/s, opposite to girl’s v
32 · Two masses of 5 kg and 10 kg are connected by a compressed spring and rest on a frictionless table. After the
spring is released, the smaller mass has a velocity of 8 m/s to the left. What is the velocity of the larger mass?
Fext = 0; Σ mivi = 0
v10 = (5 × 8/10) m/s = 4 m/s to right
33* · Figure 8-52 shows the behavior of a projectile just after it has broken up into three pieces. What was the speed of
the projectile the instant before it broke up? (a) v3 (b) v3/3 (c) v3/4 (d) 4v3 (e) (v1 + v2 + v3)/4
(c) Use p i = p f = mv3 = 4mvi.
34 · A shell of mass m and speed v explodes into two identical fragments. If the shell was moving horizontally with
respect to the earth, and one of the fragments is subsequently moving vertically with the speed v, find the velocity v′ of
the other fragment.
Use p i = p f ; p i = mv i = 1/2mv j + 1/2mv′
v′ = 2v i - v j
35 ·· In a circus act, Marcello (mass 70 kg) is shot from a cannon with a muzzle velocity of 24.00 m/s at an angle of 30o
above horizontal. His partner, Tina (mass 50 kg), stands on an elevated platform located at the top of his trajectory. He
grabs her as he flies by and the two fly off together. They land in a net at the same elevation as the cannon a
horizontal distance x away. Find x.
1. Find p i at contact; vxi = (24 cos 30o) m/s = 20.8 m/s
(70 × 20.8) kg.m/s = (120 × vxf ) kg.m/s; vxf = 12.1 m/s
2. Find the time of flight to the platform
t = (24 sin 30o)/g s = 1.22 s
3. Find x = vxit + vxf t
x = [(20.8 + 12.1)1.22] m = 40.1 m
36 ·· A block and a handgun loaded with one bullet are firmly affixed to opposite ends of a massless cart that rests on a
level frictionless air table (Figure 8-53). The mass of the handgun is mg, the mass of the block is mbk, and the mass of
the bullet is mbt. The gun is aimed so that when fired, the bullet will go into the block. When the bullet leaves the barrel
of the handgun, it has a velocity vb as measured by an observer at rest with the table. Take the fall of the bullet to be
negligible and its penetration into the block to be small. (a) What is the velocity of the cart immediately after the bullet
leaves the gun barrel? (b) What is the velocity of the cart immediately after the bullet comes to rest in the block? (c)
How far has the block moved from its initial position at the moment when the bullet comes to rest in the block?
(a) Use p b = -p cart ; vcart = -[mb/(mg + mbk)]vb .
(b) Fext = 0; p i = p f = 0; vcart = 0.
(c) vrel = vb + vcart = vb[(mg+mb+mbk)/(mg+mbk)]; time of flight = L/vrel ; d = vcart t = L[mb/(mg+mb+mbk)].
37* ·· A small object of mass m slides down a wedge of mass 2m and exits smoothly onto a frictionless table. The wedge
is initially at rest on the table. If the object is initially at rest at a height h above the table, find the velocity of the wedge
when the object leaves it.
p x = 0 = mvx - 2mV; V = 1/2vx; vx =
38 ·
2 gh ; V = gh / 2 0, directed opposite to that of the mass m.
Describe how a basketball is moving when (a) its total kinetic energy is just the energy of motion of its center of
Chapter 8
Systems of Particles and Conservation of Momentum
mass, and (b) its total kinetic energy is the energy of its motion relative to its center of mass.
(a) It has a speed v but is not spinning about its axis.
(b) It has no velocity but is spinning about its center of mass.
39 · Two bowling balls are moving with the same velocity, but one just slides down the alley, whereas the other rolls
down the alley. Which ball has more energy?
The rolling ball has more energy, 1/2mvcm2 + Krel .
40 · A 3-kg block is traveling to the right at 5 m/s, and a second 3-kg block is traveling to the left at 2 m/s. (a) Find the
total kinetic energy of the two blocks in this reference frame. (b) Find the velocity of the center of mass of the twobody system. (c) Find the velocities of the two blocks relative to the center of mass. (d) Find the kinetic energy of the
motion of the blocks relative to the center of mass. (e) Show that your answer for part (a) is greater than your answer
for part (d) by an amount equal to the kinetic energy of the center of mass.
(a) K = K1 + K2
K = 1/2[3 × 25 + 3 × 4] J = 43.5 J
(b) Use Equ. 8-13
vcm = (3 × 5 - 3 × 2)/6 i m/s = 1.5 i m/s
(c) vrel = v - vcm
v1,rel = 3.5 i m/s; v2,rel = -3.5 i m/s
(d) Krel = K1,rel + K2,rel
Krel = 1/2(3 × 3.52 + 3 × 3.52) J = 36.75 J
(e) Find Kcm
Kcm = 1/2(6 × 1.52) J = 6.75 J = K - Krel
41* · Repeat Problem 40 with the second, 3-kg block replaced by a block having a mass of 5 kg and moving to the right
at 3 m/s.
We follow the same procedures as in the preceding problem.
(a) K = 1/2(3 × 52 + 5 × 32) J = 60 J; (b) vcm = (3 × 5 + 5 × 3)/8 i m/s = 3.75 i m/s; (c) v1,rel = 1.25 m/s,
2
2
2
v2,rel = -0.75 m/s; (d) Krel = 1/2(3 × 1.25 + 5 × 0.75 ) J = 3.75 J; (e) Kcm = 1/2(8 × 3.75 ) = 56.25 J = K - Krel .
42 · Explain why a safety net can save the life of a circus performer.
It reduces the force acting on the performer by increasing ∆t. Note that ∆p = F∆t is constant.
43 · How might you estimate the collision time of a baseball and bat?
Assume that the ball travels at 80 mi/h = 35 m/s. The ball stops in a distance of about 1 cm. So the distance
traveled is about 2 cm at an average speed of about 18 m/s. The collision time is 0.02/18 = 1 ms.
44 · Why does a wine glass survive a fall onto a carpet but not onto a concrete floor?
The average force on the glass is less when falling on a carpet because ∆t is longer.
45* · A soccer ball of mass 0.43 kg leaves the foot of the kicker with an initial speed of 25 m/s. (a) What is the impulse
imparted to the ball by the kicker? (b) If the foot of the kicker is in contact with the ball for 0.008 s, what is the
average force exerted by the foot on the ball?
(a) Use Equ. 8-19
I = mv = 10.75 N.s
(b) Use Equ. 8-20
Fav = 10.75/0.008 N = 1344 N
46 · A 0.3-kg brick is dropped from a height of 8 m. It hits the ground and comes to rest. (a) What is the impulse
exerted by the ground on the brick? (b) If it takes 0.0013 s from the time the brick first touches the ground until it
comes to rest, what is the average force exerted by the ground on the brick?
Chapter 8
(a) I = ∆p
(b) Fav = I/∆t
Systems of Particles and Conservation of Momentum
I = mv = 0.3 × (2 × 9.81 × 8)1/2 N.s = 3.76 N.s
Fav = (3.76/0.0013) N = 2891 N
47 · A meteorite of mass 30.8 tonne (1 tonne = 1000 kg) is exhibited in the Hayden Planetarium in New York. Suppose
the kinetic energy of the meteorite as it hit the ground was 617 MJ. Find the impulse I experienced by the meteorite up
to the time its kinetic energy was halved (which took about t = 3.0 s). Find also the average force F exerted on the
meteorite during this time interval.
1. I = ∆p; find p i and p f ; use Equ. 8-23
p i = (2 × 30.8 × 103 × 617 × 106)1/2 kg.m/s
p f2
2
2. = p i /2
3. Fav = I/∆t
= 6.165 × 106 kg.m/s
p f = p i/ 2 0; ∆p = p i(1 - 1/
Fav = 0.602 MN
2 0) = I = 1.81 MN .s
48 ·· When a 0.15-kg baseball is hit, its velocity changes from +20 m/s to -20 m/s. (a) What is the magnitude of the
impulse delivered by the bat to the ball? (b) If the baseball is in contact with the bat for 1.3 ms, what is the average
force exerted by the bat on the ball?
(a) I = ∆p
I = 0.15 × 40 N.s = 6 N.s
(b) Fav = I/∆t
Fav = (6/1.3 × 10-3) N = 4.62 kN
49* ·· A 300-g handball moving with a speed of 5.0 m/s strikes the wall at an angle of 40o and then bounces off with the
same speed at the same angle. It is in contact with the wall for 2 ms. What is the average force exerted by the ball on
the wall?
o
1. Find ∆v; vxi = v0 cos 40o, vxf = -v0 cos 40o
∆v = 2 × 5.0 × cos 40 m/s = 7.66 m/s
2. Fav = m∆v/∆t
Fav = 0.3 × 7.66/2 × 10-3 N = 1.15 kN
50 ·· A 2000-kg car traveling at 90 km/h crashes into a concrete wall that does not give at all. (a) Estimate the time of
collision, assuming that the center of the car travels halfway to the wall with constant deceleration. (Use any reasonable length for the car.) (b) Estimate the average force exerted by the wall on the car.
(a) Assume a car length of 6 m; then ∆t = (3/2vav) s = 1.5/(25/2) s = 0.12 s.
(b) Fav = ∆p/∆t = (2000 × 25/0.12) N = 417 kN.
51 ·· You throw a 150-g ball to a height of 40 m. (a) Use a reasonable value for the distance the ball moves while it is in
your hand to calculate the average force exerted by your hand and the time the ball is in your hand while you throw it.
(b) Is it reasonable to neglect the weight of the ball while it is being thrown?
(a) Take d = 0.7 m; find v on leaving hand and vav
v = (2gh)1/2 = (2 × 9.81 × 40)1/2 = 28 m/s; vav = 14 m/s
∆t = d/vav
∆t = 0.7/14 s = 0.05 s
Fav = ∆p/∆t
Fav = (0.15 × 28/0.05) N = 84 N
(b) w/Fav = mg/Fav = 0.0175
Yes; w is less than 2% of the average force
52 ·· A handball of mass 300 g is thrown straight against a wall with a speed of 8 m/s. It rebounds with the same speed.
(a) What impulse is delivered to the wall? (b) If the ball is in contact with the wall for 0.003 s, what average force is
exerted on the wall by the ball? (c) The ball is caught by a player who brings it to rest. In the process, her hand moves
back 0.5 m. What is the impulse received by the player? (d) What average force was exerted on the player by the
Chapter 8
ball?
(a) See Problem 8-48
(b) See Problem 8-48
(c) I = m∆v
(d) ∆t = d/vav; Fav = I/∆t = Ivav/d
Systems of Particles and Conservation of Momentum
I = 0.3 × 16 N.s = 4.8 N.s
Fav = 4.8/0.003 N = 1.6 kN
I = 0.3 × 8 N.s = 2.4 N.s
Fav = (2.4 × 4/0.5) N = 19.2 N
53* ··· The great limestone caverns were formed by dripping water. (a) If water droplets of 0.03 mL fall from a height of
5 m at a rate of 10 per minute, what is the average force exerted on the limestone floor by the droplets of water? (b)
Compare this force to the weight of a water droplet.
(a) 1. Find the mass of the droplet
m = (3 × 10-5 L)(1.0 kg/L) = 3 × 10-5 kg
2. Find v at impact
v = (2 × 9.81 × 5)1/2 m/s = 9.9 m/s
3. Find Fav = Nm∆v/∆t; N/∆t = (10/60) s-1
Fav = (3 × 10-5 × 9.9/6) N = 4.95 × 10-5 N
(b) w/Fav = 3 × 9.81/4.95 = 6
w is about six times the average force due to 10 drops
54 ··· A favorite game at picnics is the egg toss. Two people toss a raw egg back and forth as they move farther apart.
If the force required to break the egg’s shell is about 5 N and the mass of the egg is 50 g, estimate the maximum
separation distance for the egg throwers. Make whatever assumptions seem reasonable.
Assume θ = 45o for the maximum range; then R = v02/g. Assume the movement of the hand is d = 0.7 m. With Fmax =
5 N, we have 5 × 0.7 J = 1/2 × 0.05 × 9.81 × R J. So R = 14.3 m.
55 · True or false: (a) In any perfectly inelastic collision, all the kinetic energy of the bodies is lost. (b) In a head-on
elastic collision, the relative speed of recession after the collision equals the relative speed of approach before the
collision. (c) Kinetic energy is conserved in an elastic collision.
(a) False (b) True (c) True
56 ·· Under what conditions can all the initial kinetic energy of colliding bodies be lost in a collision?
This occurs in a perfectly inelastic collision in which the velocity of the center of mass is zero.
57* ·· Consider a perfectly inelastic collision of two objects of equal mass. (a) Is the loss of kinetic energy greater if the
two objects have oppositely directed velocities of equal magnitude v/2, or if one of the two objects is initially at rest and
the other has an initial velocity of v? (b) In which situation is the percentage loss in kinetic energy the greatest?
(a) Case 1: Ki = 2(1/2mv2/4), Kf = 0; ∆K = mv2/4. Case 2: Ki = 1/2mv2, Kf = 1/2[2m × (v/2)2] = mv2/4; ∆K = mv2/4.
The energy losses are the same.
(b) The percentage loss is greatest (infinite) in case 1.
58 ·· A mass m1 traveling with a speed v makes a head-on elastic collision with a stationary mass m2. In which scenario
will the energy imparted to m2 be greatest? (a) m2 << m1 (b) m2 = m1 (c) m2 >> m1 (d) none of the above
(b) All of the energy is imparted to m2.
59 · Joe and Sal decide that little Ronny is well-behaved enough to sit at the table with the family for Thanksgiving
dinner. They are wrong. Ronny throws a 150-g handful of mashed potatoes horizontally with a speed of 5 m/s. It
strikes a 1.2-kg gravy boat that is initially at rest on the frictionless table. If the potatoes stick to the gravy boat, what is
the speed of the combined system as it slides down the table toward Grandpa?
Use p i = p f; vf = vim1/(m1 + m2)
vf = (5 × 0.15/1.35) m/s = 0.556 m/s
Chapter 8
Systems of Particles and Conservation of Momentum
60 · A 2000-kg car traveling to the right at 30 m/s is chasing a second car of the same mass that is traveling to the right
at 10 m/s. (a) If the two cars collide and stick together, what is their speed just after the collision? (b) What fraction of
the initial kinetic energy of the cars is lost during this collision? Where does it go?
(a) Use p i = m1v1 + m2v2 = (m1 + m2)v; m1=m2=m
vf = m(v1 + v2)/2m = 20 m/s
2
2
2
2
2
2
2
2
(b) Ki = 1/2m(v1 + v2 ); Kf =mvf
∆K/Ki = (v1 + v2 - 2vf )/(v1 + v2 ) = 0.2
The energy goes into heat, sound, and
the deformation of metal.
61* · An 85-kg running back moving at 7 m/s makes a perfectly inelastic collision with a 105-kg linebacker who is
initially at rest. What is the speed of the players just after their collision?
Use p i = p f; vf = vim1/(m1 + m2)
vf = 85 × 7/190 = 3.13 m/s
62 · A 5.0-kg object with a speed of 4.0 m/s collides head-on with a 10-kg object moving toward it with a speed of 3.0
m/s. The 10-kg object stops dead after the collision. (a) What is the final speed of the 5-kg object? (b) Is the collision
elastic?
(a) p i = p f
vf,5 = (10 × 3.0 - 5 × 4.0)/5 m/s = 2 m/s
(b) If Ki = Kf, the collision is elastic
Ki = (40 + 45) J; Kf = 10 J; inelastic collision
63 · A ball of mass m moves with speed v to the right towards a much heavier bat that is moving to the left with speed
v. Find the speed of the ball after it makes an elastic collision with the bat.
Use Equ. 8-29 and the fact that vi,bat ≈ vf,bat ; take left
vf,ball = vi,ball + 2vi,bat
as the positive direction
vf,ball = 3v
64 ·· During the Great Muffin Wars of ‘98, students from rival residences became familiar with the characteristics of
various muffins. Mushy Pumpkin Surprise, for example, was good for temporarily blinding an attacker, while Mrs.
O’Brien’s Bran Muffins, having the density of lacrosse balls, were used more sparingly, and mainly as a deterrent.
According to the rules, all muffins must have a mass of 0.3 kg. During one of the more memorable battles, a muffin
moving to the right at 5 m/s collides with a muffin moving to the left at 2 m/s. Find the final velocities if (a) it is a
perfectly inelastic collision of two pumpkin muffins and (b) it is an elastic collision of two bran muffins.
(a) Use p i = p f; vf = (v1im1+v2im2)/(m1 + m2)
vf = m(5 - 2)/2m m/s = 1.5 m/s, to the right
(b) 1. Find vcm and transform to CM system
vcm = 1.5 m/s; u 1i = 3.5 m/s, u 2i = -3.5 m/s
2. Use u f = -u i and transform back
u 1f = -3.5 m/s, u 2f = 3.5 m/s; v1f = -2 m/s, v2f = 5 m/s
65* ·· Repeat Problem 64 with a second (illegal) muffin having a mass of 0.5 kg and moving to the right at 3 m/s.
(a) Use p i = p f; vf = (v1im1+v2im2)/(m1 + m2)
vf = [(0.3 × 5 + 0.5 × 3)/0.8] m/s = 3.75 m/s
(b) 1. Transform to CM system; u = v - vcm
vcm = 3.75 m/s; u 1i = 1.25 m/s, u 2i = -0.75 m/s
2. Use u f = -u i; transform back to lab system
u 1f = -1.25 m/s, u 2f = 0.75 m/s; v1f = 2.5 m/s,
v2f = 4.5 m/s
66 ·· A proton of mass m undergoes a head-on elastic collision with a stationary carbon nucleus of mass 12m. The
speed of the proton is 300 m/s. (a) Find the velocity of the center of mass of the system. (b) Find the velocity of the
proton after the collision.
(a) Use Equ. 8-13
vcm = 300m/13m m/s = 23.1 m/s
Chapter 8
(b) Use Equ. 8-30a
Systems of Particles and Conservation of Momentum
vpf = [(-11/13)300] m/s = -254 m/s
67 ·· A 3-kg block moving at 4 m/s makes an elastic collision with a stationary block of mass 2 kg. Use conservation of
momentum and the fact that the relative velocity of recession equals the relative velocity of approach to find the
velocity of each block after the collision. Check your answer by calculating the initial and final kinetic energies of each
block.
Equs. 8-30a and 8-30b are a direct consequence of momentum conservation and Equ. 8-29. (see Example 8-17)
1. Use Equs. 8-30a and 8-30b
v3f = (1 × 4/5) m/s = 0.8 m/s; v2f = (6 × 4/5) m/s
= 4.8 m/s
2. Evaluate Ki and Kf
Ki = 1.5 × 16 J = 24 J; Kf = (1.5 × 0.82+1 × 4.82) J = 24 J
68 ·· Night after night, Lucy is tormented by nocturnal wailing from the house next door. One day she seizes a revolver,
stalks to the neighbor’s window with a crazed look in her eye, takes aim, and fires a 10-g bullet into her target: a 1.2 kg
saxophone that rests on a frictionless surface. The bullet passes right through and emerges on the other side with a
speed of 100 m/s, and the saxophone is given a speed of 4 m/s. Find the initial speed of the bullet, and the amount of
energy dissipated in its trip through the saxophone.
1. Use p i = p f
(0.01vi) kg.m/s = (0.01 × 100 + 1.2 × 4) kg.m/s;
2. Find ∆K = Ki - Kf
vi = 580 m/s
2
2
2
∆K = [0.005 × 580 - 0.6 × 4 - 0.005 × 100 ] J
= 1622 J
69* ·· A block of mass m1 = 2 kg slides along a frictionless table with a speed of 10 m/s. Directly in front of it, and
moving in the same direction with a speed of 3 m/s, is a block of mass m2 = 5 kg. A massless spring with spring
constant k = 1120 N/m is attached to the second block as in Figure 8-54. (a) Before m1 runs into the spring, what is
the velocity of the center of mass of the system? (b) After the collision, the spring is compressed by a maximum
amount ∆x. What is the value of ∆x? (c) The blocks will eventually separate again. What are the final velocities of the
two blocks measured in the reference frame of the table?
(a) Use Equ. 8-13
vcm = (2 × 10 + 5 × 3)/7 m/s = 5 m/s
(b) 1. At max. compression, u 1 = u 2 = 0; K = Kcm
Kcm = 1/2 × 7 × 52 J = 87.5 J
2. Use conservation of energy: 1/2k ∆x2 = Ki - Kcm
Ki = (100 + 22.5) J = 122.5 J; 1/2k ∆x2 = 35 J
3. Solve for and evaluate ∆x
∆x = (2 × 35/1120) 1/2 m = 0.25 m = 25 cm
(c) 1. Collision is elastic; find u 1i and u 2i; u 1f and u 2f
u 1i = 5 m/s, u 2i = -2 m/s; u 1f = -5 m/s, u 2f = 2 m/s
2. Transform to reference frame of table
v1f = 0 m/s, v2f = 7 m/s
70 ·· A bullet of mass m is fired vertically from below into a block of wood of mass M that is initially at rest, supported
by a thin sheet of paper. The bullet blasts through the block, which rises to a height H above its initial position before
falling back down. The bullet continues rising to a height h. (a) Express the upward velocity of the bullet and the block
immediately after the bullet exits the block in terms of h and H. (b) Use conservation of momentum to express the
speed of the bullet before it enters the block of wood in terms of given parameters. (c) Obtain expressions for the
mechanical energies of the system before and after the inelastic collision. (d) Express the energy dissipated in the
Chapter 8
Systems of Particles and Conservation of Momentum
block of wood in terms of m, h, M, and H.
(a) vm =
2 gh 0; vM =
2 gH 0.
(b) mvmi = mvm + MvM; vmi =
2 gh + (M/m) 2 gH .
(c) Ei = 1/2mvmi2 = mg[h = 2(M/m)
hH + (M/m )2H]; Ef = g(mh + MH).
(d) Wf = Ei - Ef = gMH[2 h / H + (M/m) – 1].
71 ·· A proton of mass m is moving with initial speed v0 toward an α particle of mass 4m, which is initially at rest.
Because both particles carry positive electrical charge, they repel each other. Find the speed v′ of the α particle (a)
when the distance between the two particles is least, and (b) when the two particles are far apart.
(a) This problem is similar to Problem 8-69; here the electrostatic repulsion takes the place of the spring. When
the distance between the two particles is least, both move at the same speed, namely vcm.
Find vcm
vcm = vα′ = v0/5 = 0.2v0
(b) Use Equ. 8-30b
vαf = (2/5)v0 = 0.4v0
72 ·· A 16-g bullet is fired into the bob of a ballistic pendulum of mass 1.5 kg. When the bob is at its maximum height,
the strings make an angle of 60o with the vertical. The length of the pendulum is 2.3 m. Find the speed of the bullet.
1. Use p i = p f
0.016v1 = 1.516v2
v2 = [2 × 9.81 × 2.3 × (1 - cos 60o)]1/2
2. From energy conservation, v2 = 2 gH 0
v2 = 4.75 m/s, v1= 450 m/s
3. Evaluate v2 and v1, the speed of the bullet
73* ·· A bullet of mass m1 is fired with a speed v into the bob of a ballistic pendulum of mass m2. The bob is attached to a
very light rod of length L that is pivoted at the other end. The bullet is stopped in the bob. Find the minimum v such that
the bob will swing through a complete circle.
2
1. Find vi of bob + bullet to make complete circle
1/2(m1+m2)vi = 2gL(m1+m2); vi = 2 gL 0
2. Use p i = p f to find v
v = 2[(m1+m2)/ m1)] gL 0
74 ·· A bullet of mass m1 is fired with a speed v into the bob of a ballistic pendulum of mass m2. Find the maximum
height h attained by the bob if the bullet passes through the bob and emerges with a speed v/2.
1. Use p i = p f to find v2
m1v = 1/2m1v + m2v2; v2 = 1/2(m1/m2)v
2
2
2
2
2. Use conservation of energy to find h
1/2m2v2 = m2gh; h = (m1v/2m2) /2g = (v /8g)(m1/m2)
75 ·· A 3-kg bomb slides along a frictionless horizontal plane in the x direction at 6 m/s. It explodes into two pieces, one
of mass 2 kg and the other of mass 1 kg. The 1-kg piece moves along the horizontal plane in the y direction at 4 m/s.
(a) Find the velocity of the 2-kg piece. (b) What is the velocity of the center of mass after the explosion?
We shall do part (b) first and use the result to solve part (a).
(b) vcm = vi from conservation of p
vcm = 6 m/s i
(a) Use p i = p f = (3 kg) × vcm
18 m/s i = 1 × 4 m/s j + 2 × v2; v2 = 9 m/s i - 2 m/s j
76 ··
The beryllium isotope 4Be is unstable and decays into two α particles (helium nuclei of mass m = 6.68 × 10-27 kg)
Chapter 8
Systems of Particles and Conservation of Momentum
with the release of 1.5 × 10-14 J of energy. Determine the velocities of the two α particles that arise from the decay of
a 4Be nucleus at rest.
From momentum conservation it follows that the velocities of the two α particles are equal in magnitude and
oppositely directed.
Use energy conservation; solve for vα
2(1/2mαvα2) = 1.5 × 10-14 J; vα = 1.5 × 106 m/s
77* ·· The light isotope of lithium, 5Li, is unstable and breaks up spontaneously into a proton (hydrogen nucleus) and an α
particle (helium nucleus). In this process, a total energy of 3.15 × 10-13 J is released, appearing as the kinetic energy of
the two reaction products. Determine the velocities of the proton and α particle that arise from the decay of a 5Li
nucleus at rest. (Note: The masses of the proton and alpha particle are mp = 1.67 × 10-27 kg and mα = 4mp = 6.68 ×
10-27 kg.)
1. Use p i = p f = 0
4mpvα = mpvp; vα = vp/4
2
2
2
-13
2. Use energy conservation
1/2mpvp + 1/2mαvα = (5/8)mpvp = 3.15 × 10 J
3. Solve for vp and vα
vp = 1.74 × 107 m/s; vα = 4.34 × 106 m/s
78 ·· Jay and Dave decide that the best way to protest the opening of a new incinerator is to launch a stink bomb into
the middle of the ceremony. They calculate that a 6-kg projectile launched with an initial speed of 40 m/s at an angle of
30o will do the trick. The bomb will explode on impact, no one will get hurt, but everyone will stink. Perfect. However,
at the top of its flight, the bomb explodes into two fragments, each having a horizontal trajectory. To top it off—this
really isn’t their day—the 2-kg fragment lands right at the feet of Dave and Jay. (a) Where does the 4-kg fragment
land? (b) Find the energy of the explosion by comparing the kinetic energy of the projectiles just before and just after
the explosion.
(a) 1. Find h at explosion
Mgh = 1/2Mvyi2; h = (40 sin 30o)2/2g m = 20.39 m
2. Find vx at explosion
vx = 40 cos 30o m/s = 34.64 m/s
3. To land at origin, vx of 2 kg = -34.64 m/s
p i = p f; 6vx = 4v4x - 2vx; v4x = 8vx/4 = 69.3 m/s
4. Find time of flight
t = 2vyi/g = 40/9.81 s = 4.08 s
5. Find distance traveled by 4 kg
d = (2.04 × 34.64 + 2.04 × 69.3) m = 212 m
(b) 1. Find Ki and Kf at explosion site
Ki = 3 × 34.642 J = 3600 J; Kf = (34.642 + 2 × 69.32) J
= 10800 J
2. Energy of explosion = Kf - Ki
Energy of explosion = 7200 J
79 ·· A projectile of mass m = 3 kg is fired with initial speed of 120 m/s at an angle of 30o with the horizontal. At the top
of its trajectory, the projectile explodes into two fragments of masses 1 kg and 2 kg. The 2-kg fragment lands on the
ground directly below the point of explosion 3.6 s after the explosion. (a) Determine the velocity of the 1-kg fragment
immediately after the explosion. (b) Find the distance between the point of firing and the point at which the 1-kg
fragment strikes the ground. (c) Determine the energy released in the explosion.
o
(a) 1. Find v and r at point of explosion
v = vx0 i; vx0 = (120 cos 30 ) m/s = 104 m/s
y = vy02/2g = 602/19.62 m = 183.5 m
Find t1, time until explosion and x
t1 = vy0/g = 6.12 s; x = vx0t1 = 636.5 m
2. Find v2 of 2 kg after explosion
vx2 = 0;183.5 - vy2 × 3.6 - 1/2 × 9.81 × 3.62 = 0;
vy2 = -33.3 m/s
Chapter 8
Systems of Particles and Conservation of Momentum
3. Use p i = p f to find vx1 and vy1
(b) 1. Find t2, time for 1 kg to reach ground
2. Find distance traveled by 1 kg mass
(c) 1. Find Ki and Kf at explosion;
Eexpl. = ∆K = Kf - Ki
vx1 = 3 × 104 m/s = 312 m/s;
vy1 = 2 × 33.3 m/s = 66.6 m/s
2
1/2gt2 - 66.6t2 - 183.5 = 0; t2 = 15.94 s
d = (636.5 + 312 × 15.94) m = 5610 m
Ki = 1.5 × 1042 J = 16.2 kJ;
Kf = 33.32 + 1/2(3122 + 66.62) J = 52 kJ
∆K = Eexpl. = 35.8 kJ
80 ··· The boron isotope 9B is unstable and disintegrates into a proton and two α particles. The total energy released as
kinetic energy of the decay products is 4.4 × 10-14 J. In one such event, with the 9B nucleus at rest prior to decay, the
velocity of the proton is measured to be 6.0 × 106 m/s. If the two α particles have equal energies, find the magnitude
and the direction of their velocities with respect to that of the proton.
1. Show a sketch of the velocities. We assume that the
proton’s velocity is in the negative x direction. Note that
mα = 4mp = 6.68 × 10-27 kg (see Problem 8-77)
2. Use conservation of energy to find vα
3. Σ p i = 0
4. Give the angle with respect to vp
2
+ 2(1/2mαvα2) = 4.4 × 10-14 J; mαvα2 = 1.4 × 10-14 J
vα = 1.45 × 106 m/s
2(4mp)(1.45 × 106 cos θ) = mp × 6 × 106; θ = ±58.9o
o
o
o
θ′ = ±(180 - 58.9 ) = ±121.1
1/2mpvp
81* · The coefficient of restitution for steel on steel is measured by dropping a steel ball onto a steel plate that is rigidly
attached to the earth. If the ball is dropped from a height of 3 m and rebounds to a height of 2.5 m, what is the
coefficient of restitution?
Find the ratio vrec/vapp and use Equ. 8-31
2 ghrec
v rec
2.5
e0 =
=
=
= 0.913
2
gh
3
.0
vapp
app
82 · According to the official rules of racquetball, a ball acceptable for tournament play must bounce to a height of
between 173 and 183 cm when dropped from a height of 254 cm at room temperature. What is the acceptable range
Chapter 8
Systems of Particles and Conservation of Momentum
of values for the coefficient of restitution for the racquetball–floor system?
See Problem 8-81
emin = 173 / 254 = 0.825; emax =
183 / 254 = 0.8490
83 · A ball bounces to 80% of its original height. (a) What fraction of its mechanical energy is lost each time it
bounces? (b) What is the coefficient of restitution of the ball–floor system?
(a) Since K ∝ h, 20% of energy is lost with each bounce.
(b) Since K ∝ v2, e = (0.8) 1/2 = 0.894.
84 ·· A 2-kg object moving at 6 m/s collides with a 4-kg object that is initially at rest. After the collision, the 2-kg object
moves backward at 1 m/s. (a) Find the velocity of the 4-kg object after the collision. (b) Find the energy lost in the
collision. (c) What is the coefficient of restitution for this collision?
(a) Use p i = p f
(2 × 6) kg.m/s = [-(2 × 1) + 4 × v] kg.m/s; v = 3.5 m/s
(b) Eloss = Ki - Kf
Eloss = 36 J - (1 + 2 × 3.52) J = 10.5 J
(c) Use Equ. 8-31
e = (3.5 + 1)/6 = 0.75
85* ·· A 2-kg block moving to the right with speed 5 m/s collides with a 3-kg block that is moving in the same direction at
2 m/s, as in Figure 8-55. After the collision, the 3-kg block moves at 4.2 m/s. Find (a) the velocity of the 2-kg block
after the collision, and (b) the coefficient of restitution for the collision.
(a) Use p i = p f
(2 × 5 + 3 × 2) kg.m/s = (2v1f + 3 × 4.2) kg.m/s;
(b) Use Equ. 8-31
v1f = 1.7 m/s
e = (4.2 - 1.7)/(5 - 2) = 0.833
86 · In a pool game, the cue ball, which has an initial speed of 5 m/s, makes an elastic collision with the eight ball, which
is initially at rest. After the collision, the eight ball moves at an angle of 30o with the original direction of the cue ball.
(a) Find the direction of motion of the cue ball after the collision. (b) Find the speed of each ball. Assume that the balls
have equal mass.
(a) Elastic collision, use energy conservation;
vci2 = vcf2 + v82; Note that vci, vcf, and v8 form right triangle
m1=m2
o
o
θcf + θ8 = 90
θcf = 60
(b) 1. Use p i = p f
5 m/s = vcf cos 60o + v8 cos 30o; vcf sin 60o = v8 sin 30o
2. Solve for vcf and v8
vcf = 2.5 m/s; v8 = 4.33 m/s
87 ·· An object of mass M 1 = m collides with velocity v0 i into an object of mass M 2 = 2m with velocity 1/2v0 j. Following
the collision, the mass m2 has a velocity v0/4 i. (a) Determine the velocity of the mass m1 after the collision. (b) Was
this an elastic collision? If not, express the energy change in terms of m and v0.
(a) 1. Use p i = p f
p i = mv0 i + mv0 j = p f = 1/2mv0 i + mv1f
2
2
2. Find v1f
v1f = v0(1/2 i + j); v1f = (5/4)v0
2
2
2
2
2
(b) Find ∆E = Ki - Kf
∆E = 1/2mv0 + mv0 /4 - mv0 /16 - (5/8)mv0 = mv0 /16
Chapter 8
Systems of Particles and Conservation of Momentum
88 ·· A puck of mass 0.5 kg approaches a second, similar puck that is stationary on frictionless ice. The initial speed of
the moving puck is 2 m/s. After the collision, one puck leaves with a speed v1 at 30o to the original line of motion; the
second puck leaves with speed v2 at 60o, as in Figure 8-56. (a) Calculate v1 and v2. (b) Was the collision elastic?
(b) Yes; this is identical to Problem 8-86. Since the angle between pucks is 90o, the collision is elastic.
(a) Follow procedure of Problem 8-86(b) with vci = 2 m/s. v1 = 1 m/s, v2 = 1.73 m/s.
89* ·· Figure 8-57 shows the result of a collision between two objects of unequal mass. (a) Find the speed v2 of the
larger mass after the collision and the angle θ2. (b) Show that the collision is elastic.
(a) 1. Use p i = p f ;
3mv0 = 5 0mv0cos θ1 + 2mv2cos θ2;
Note: 5 0sin θ1 = 2, 5 0cos θ1 = 1
5 0mv0 sin θ1 = 2mv2 sin θ2
2. Solve for θ2
2 cot θ2 = 3 - 2; cot θ2 = 1, θ2 = 45o
3. Solve for v2
v2= 2 0v0
(b) Find Ki and Kf
Ki = 4.5mv02; Kf = 2.5mv02 + 2mv02 = 4.5mv02; Q.E.D.
90 ·· A ball moving at 10 m/s makes an off-center elastic collision with another ball of equal mass that is initially at rest.
The incoming ball is deflected at an angle of 30o from its original direction of motion. Find the velocity of each ball
after the collision.
This is identical to Problem 8-88 except that the initial speed of the object is 10 m/s. It follows that the velocities
are 5 m/s at 60o and 8.66 m/s at 30o.
91 ··· A particle has an initial speed v0. It collides with a second particle that is at rest and is deflected through an angle
φ . Its speed after the collision is v. The second particle recoils. Its velocity makes an angle θ with the initial direction
of the first particle. (a) Show that tan θ = (v sin φ )/(v0 - v cos φ ). (b) Do you have to assume that the collision is
either elastic or inelastic to get the result in part (a)?
(a) Let m1 and m2 be the incoming and struck particles; assume v0 is in the x direction. From p i = p f, one obtains
m1v0 = m1v cos φ + m2v2 cos θ (1) and m1v sin φ = m2v2 sin θ (2). m2v2 cos θ = m1v0 - m1v cos φ (1a).
Divide (2) by (1a) to obtain: tan θ = (v sin φ )/(v0 - v cos φ ).
(b) You did not use energy conservation. Therefore, the result is valid for elastic and inelastic collisions.
92 · Describe a perfectly inelastic collision as viewed in the center-of-mass reference frame.
The two objects approach with equal but opposite momenta and remain at rest after the collision.
93* ·· A particle with momentum p 1 in one dimension makes an elastic collision with a second particle of momentum p 2 =
-p 1 in the center-of-mass reference frame. After the collision its momentum is p 1'. Write the total initial and final
energies in terms of p 1 and p 1', and show that p 1' = ±p 1. If p 1' = -p 1, the particle is merely turned around by the collision and leaves with the speed it had initially. What is the significance of the plus sign in your solution?
Krel = p 12/2m1 + p 12/2m2 = p 12(m1 + m2)/2m1m2 ; Kcm = (2p 1)2/2(m1 + m2) = 2p 12/(m1 + m2); K = Krel + Kcm , i.e.,
K = (p 12/2)[(m12 + 6m1m2 + m22)/(m12m2 + m1m22)]. In an elastic collision, Ki = Kf. Consequently, (p 1')2 = (p 1)2 and
p 1' = ±p 1. If p 1' = p 1, the particles do not collide.
94 ·· A 3-kg block is traveling to the right at 5 m/s, and a 1-kg block is traveling to the left at 3 m/s. (a) Find the velocity
vcm of the center of mass. (b) Subtract vcm from the velocity of each block to find the velocity of each block in the
center-of-mass reference frame. (c) After they make an elastic collision, the velocity of each block is reversed in this
frame. Find the velocity of each block after an elastic collision. (d) Transform back into the original frame by adding
Chapter 8
Systems of Particles and Conservation of Momentum
vcm to the velocity of each block. (e) Check your result by finding the initial and final kinetic energies of the blocks in
the original frame.
(a) Use Equ. 8-13
vcm = [(15 - 3)/4] m/s = 3 m/s
(b) Follow the indicated procedure
u 3 = 2 m/s, u 1 = -6 m/s
(c) Follow procedure
u 3′ = -2 m/s, u 1′ = 6 m/s
(d) Transform back to lab system
v3′ = 1 m/s, v1′ = 9 m/s
(e) Evaluate Ki and Kf
Ki = 1/2(3 × 25 + 1 × 9) = 42 J; Kf = 1/2(3 × 1 + 1 × 81)
= 42 J
95 ·· Repeat Problem 94 with a second block having a mass of 5 kg and moving to the right at 3 m/s.
Following same procedures one obtains:
(a) vcm = 3.75 m/s. (b) u 3 = 1.25 m/s, u 5 = -0.75 m/s. (c) u 3′ = -1.25 m/s, u 5′ = 0.75 m/s.
(d) v3′ = 2.5 m/s, v5′ = 4.5 m/s. (e) Ki = 1/2(3 × 5 + 5 × 9) J = 60 J; Kf = 1/2(3 × 6.25 + 5 × 20.25) J = 60 J.
96 ·· A rocket burns fuel at a rate of 200 kg/s and exhausts the gas at a relative speed of 6 km/s. Find the thrust of the
rocket.
Use Equ. 8-39
Fth = 6 × 103 × 200 = 1.2 MN
97* ·· The payload of a rocket is 5% of its total mass, the rest being fuel. If the rocket starts from rest and moves with
no external forces acting on it, what is its final velocity if the exhaust velocity of its gas is 5 km/s?
1. No external forces are acting
Equ. 8-42 reduces to vf = -u ex ln(mf /m0)
2. Evaluate vf for mf /m0 = 1/20
vf = [5 ln(20)] km/s = 15 km/s
98 ·· A rocket moves in free space with no external forces acting on it. It starts from rest and has an exhaust speed of
3 km/s. Find the final velocity if the payload is (a) 20%, (b) 10%, (c) 1%.
From Problem 8-97, vf = u ex ln(m0/mf).
(a) m0/mf = 5; evaluate vf
vf = 3 ln(5) km/s = 4.83 km/s
(b) m0/mf = 10
vf = 3 ln(10) km/s = 6.91 km/s
(c) m0/mf = 100
vf = 3 ln(100) km/s = 13.8 km/s
99 ·· A rocket has an initial mass of 30,000 kg, of which 20% is the payload. It burns fuel at a rate of 200 kg/s and
exhausts its gas at a relative speed of 1.8 km/s. Find (a) the thrust of the rocket, (b) the time until burnout, and (c) its
final speed assuming it moves upward near the surface of the earth where the gravitational field g is constant.
(a) Use Equ. 8-39
Fth = 1.8 × 103 × 200 N = 360 kN
(b) mass of fuel = 0.8 × 30,000 kg = 24,000 kg
t = 24,000/200 s = 120 s
(c) Use Equ. 8-42
vf = [1.8 × 103 ln(5) - 9.81 × 120] m/s = 1.72 km/s
100 · Why can friction and the force of gravity usually be neglected in collision problems?
The event takes place in such a short time interval that the effects of gravity or friction are negligible.
101* · The condition necessary for the conservation of momentum of a given system is that (a) energy is
conserved. (b) one object is at rest. (c) no external force acts. (d) internal forces equal external forces. (e) the net
(e)
102 · As a pendulum bob swings back and forth, is the momentum of the bob conserved? Explain why or why not.
No; the net external force is the sum of the force of gravity and the tension in the string; these do not add to
Chapter 8
Systems of Particles and Conservation of Momentum
zero.
103 · A model-train car of mass 250 g traveling with a speed of 0.50 m/s links up with another car of mass 400 g that is
initially at rest. What is the speed of the cars immediately after they have linked together? Find the initial and final
kinetic energies.
1. Use p i = p f
vf = [(250 × 0.5)/650] m/s = 0.192 m/s
2. Find Ki and Kf
KI = 1/2(0.25 × 0.52) J = 31.25 mJ;Kf = 1/2(0.65 × 0.1922) =
12 mJ
104 · (a) Find the total kinetic energy of the two model-train cars of Problem 103 before they couple. (b) Find the initial
velocities of the two cars relative to the center of mass of the system, and use them to calculate the initial kinetic
energy of the system relative to the center of mass. (c) Find the kinetic energy of the center of mass. (d) Compare
your answers for (b) and (c) with that for (a).
(a) See Problem 8-103
Ki = 31.25 mJ
(b) 1. See Problem 8-103; Use Equ. 8-34
vcm = 0.192 m/s; u 1 = 0.308 m/s, u 2 = -0.192 m/s
2. Find Ki,rel
Ki,rel = 1/2(0.25 × 0.3082 + 0.4 × 0.1922) J = 19.23 mJ
(c) Find Kcm = 1/2Mvcm2
Kcm = 1/2(0.65 × 0.1922) J = 12 mJ
(d) Ki = Ki,rel + Kcm
105* · A 4-kg fish is swimming at 1.5 m/s to the right. He swallows a 1.2-kg fish swimming toward him at 3 m/s.
Neglecting water resistance, what is the velocity of the larger fish immediately after his lunch?
Use Equ. 8-13
v = vcm = (4 × 1.5 - 1.2 × 3)/5.2 m/s = 0.462 m/s
106 · A 3-kg block moves at 6 m/s to the right while a 6-kg block moves at 3 m/s to the right. Find (a) the total kinetic
energy of the two-block system, (b) the velocity of the center of mass, (c) the center-of-mass kinetic energy, and (d)
the kinetic energy relative to the center of mass.
(a) Kt = K1 + K2
Kt = 1/2(3 × 62 + 6 × 32) J = 81 J
(b) Use Equ. 8-13
vcm = (3 × 6 + 6 × 3)/9 m/s = 4 m/s
(c) Kcm = 1/2Mvcm2
(d) Krel = Kt - Kcm (see Problem 8-104)
Kcm = 1/2(9 × 42) J = 72 J
Krel = 9 J
107 · A 1500-kg car traveling north at 70 km/h collides at an intersection with a 2000-kg car traveling west at
55 km/h. The two cars stick together. (a) What is the total momentum of the system before the collision? (b)
Find the magnitude and direction of the velocity of the wreckage just after the collision.
5
5
(a) p = p 1 + p 2
p = 1.05 × 10 j kg.km/h - 1.1 × 10 i kg.km/h
(b) vf = vcm = p /M
vf = (1.052+1.12)1/2 × 105/3500 km/h = 43.4 km/h
-1
o
θ = tan (1.1/1.05) = 46.3 west of north
108 ·
The great white shark can have a mass as great as 3000 kg. Suppose such a shark is cruising the ocean when it
Chapter 8
Systems of Particles and Conservation of Momentum
spots a meal below it: a 200.0-kg fish swimming horizontally at 8.00 m/s. The shark rushes vertically downward at
3.00 m/s and swallows the prey at once. At what angle to the vertical θ will the shark be moving immediately after
the snack? What is the final speed of the shark? (Neglect any drag effects of the water.)
-1
o
1. θ = tan-1(p x/p y)
θ = tan [(200 × 8)/(3000 × 3)] = 10.1
2. v = vcm = (1/M)(p x2 + p y2)1/2
v = (1/3200)(90002 + 16002)1/2 m/s = 2.86 m/s
109* · Repeat Problem 106 for a 3-kg block moving at 6 m/s to the right and a 6-kg block moving at 3 m/s to the left.
Follow the procedures outlined in Problem 8-106. One obtains the following results.
(a) Kt = 81 J (b) vcm = 0 (c) Kcm = 0 (d) Krel = 81 J
110 · Repeat Problem 106 for a 3-kg block moving at 10 m/s to the right and a 6-kg block moving at 1 m/s to the right.
Follow the procedures outlined in Problem 8-106. One obtains the following results.
(a) Kt = 153 J. (b) vcm = 4 m/s. (c) Kcm = 72 J. (d) Krel = 81 J.
111 ·· A 60-kg woman stands on the back of a 6-m-long, 120-kg raft that is floating at rest in still water with no friction.
The raft is 0.5 m from a fixed pier, as in Figure 8-58. (a) The woman walks to the front of the raft and stops. How
far is the raft from the pier now? (b) While the woman walks, she maintains a constant speed of
3 m/s relative to the raft. Find the total kinetic energy of the system (woman plus raft), and compare with the
kinetic energy if the woman walked at 3 m/s on a raft tied to the pier. (c) Where does this energy come from,
and where does it go when the woman stops at the front of the raft? (d) On land, the woman can put a lead shot 6 m.
She stands at the back of the raft, aims forward, and puts the shot so that just after it leaves her hand, it has the same
velocity relative to her as it does when she throws it from the ground. Where does the shot land?
We shall use the following convention: Take the origin at initial position of right hand end of raft; measure positive
displacement to the left; let ∆x be the displacement of raft.
(a) Fext = 0; consquently xcm does not change
(120 × 3 + 60 × 6) kg.m
Solve for ∆x; distance to dock, d = (∆x + 0.5) m
(b) Find u w, u r, and vcm; note that the time elapsed
is 2 s
Determine Kt and K on land
= [120 × (3 + ∆x) + 60 × ∆x] kg.m
∆x = 2.0 m; d = 2.5 m
u w = -3 m/s; u r = ∆x/∆t = 1 m/s
Kt = 1/2(60 × 32 + 120) J = 330 J; K (on land) = 270 J
(c) 60 J derives from the chemical energy of
the woman.
(d) The shot will land in the water; the raft shifts
to the left.
112 ·· A 1-kg steel ball and a 2-m cord of negligible mass make up a simple pendulum that can pivot without friction
about the point O, as in Figure 8-59. This pendulum is released from rest in a horizontal position and when the ball is at
its lowest point it strikes a 1-kg block sitting at rest on a rough shelf. Assume that the collision is perfectly elastic and
take the coefficient of friction between the block and shelf to be 0.1. (a) What is the velocity of the block just after
impact? (b) How far does the block move before coming to rest?
(a) Find vball at impact
vball = (2gh)1/2 = (4 × 9.81) 1/2 m/s = 6.26 m/s
Since mball = mblock, vblock = vball after collision
vblock = 6.26 m/s
Chapter 8
Systems of Particles and Conservation of Momentum
(b) a block = µkg = 0.981 m/s2; v2 = 2as
s = 6.262/1.962 m = 20 m
113* ·· In World War I, the most awesome weapons of war were huge cannons mounted on railcars. Figure 8-60 shows
such a cannon, mounted so that it will project a shell at an angle of 30o. With the car initially at rest, the cannon fires a 200kg projectile at 125 m/s. Now consider a system composed of a cannon, shell, and railcar, all rolling on the track without
frictional losses. (a) Will the total vector momentum of that system be the same (i.e., “conserved”) before and after the
shell is fired? Explain your answer in a few words. (b) If the mass of the railcar plus cannon is 5000 kg, what will be the
recoil velocity of the car along the track after the firing? (c) The shell is observed to rise to a maximum height of 180 m as
it moves through its trajectory. At this point, its speed is 80 m/s. On the basis of this information, calculate the amount of
thermal energy produced by air friction on the shell on its way from firing to this maximum height.
(a) Momentum of system is not conserved; there is an external force, the vertical reaction force of rails.
(b) Use p xi = p xf
200 × 125 cos 30o = 5000vrec; vrec = 4.33 m/s
(c) 1. Find h without air friction; h = vy2/2g
h = 199 m
2
2
2. Wf = mg∆h + 1/2m(vx0 - vx )
Wf = 200 × 9.81 × 19 + 100[(125 cos 30o)2 - 802)] J
= 569 kJ
114 ·· A 15-g bullet traveling at 500 m/s strikes an 0.8-kg block of wood that is balanced on a table edge 0.8 m above
ground (Figure 8-61). If the bullet buries itself in the block, find the distance D at which the block hits the floor.
1. Find vcm
vcm = (500 × 0.015/0.815) m/s = 9.2 m/s
2. Find t, time to drop 0.8 m; D = vcm t
t = (1.6/9.81) 1/2 s = 0.404 s; D = 0.404 × 9.2 m = 3.72 m
115 ·· In hand-pumped railcar races, a speed of 32 km/h has been achieved by teams of four. A car of mass 350 kg is
moving at that speed toward a river when Carlos, the chief pumper, notices that the bridge ahead is out. All four
people (of mass 75 kg each) jump simultaneously backward off the car with a velocity that has a horizontal
component of 4 m/s relative to the car after jumping. The car proceeds off the bank and falls in the water a
distance 25.0 m off the bank. (a) Estimate the time of the fall of the railcar. (b) What happens to the team of
pumpers?
(a) 1. Use momentum conservation
u car × 350 = -(4 × 75)u p; u p = -(350/300)u car
2. Given that u car - u p = 4 m/s; solve for u car
u car = [4/(1 + 350/300)] m/s = 1.85 m/s
3. Use Equ. 8-34 to obtain vcar
vcar = (1.85 + 8.89) m/s = 10.74 m/s
4. t = d/vcar
t = (25/10.74) s = 2.3 s
(b) Pumpers hit the ground at 6.74 m/s = 24 km/h
They may get bruised a bit
116 ·· A constant force F = 12 N i is applied to the 8-kg mass of Problem 5 at t = 0. (a) What is the velocity of the
center of mass of the three-particle system at t = 5 s? (b) What is the location of the center of mass at t = 5 s?
(a) a cm = F/M; vcm(t) = a cm t
vcm(5) = 5 × 12/11 m/s i = 5.45 m/s i
(b) xcm(t) = xcm(0) + 1/2vcm(t) × t
xcm(5) = [(37/11) + 1/2(60/11) × 5] m = 17 m
117* ·· Two particles of mass m and 4m are moving in a vacuum at right angles as in Figure 8-62. A force F acts on both
particles for a time T. As a result, the velocity of the particle m is 4v in its original direction. Find the new velocity v′
Chapter 8
Systems of Particles and Conservation of Momentum
of the particle of mass 4m.
1. Determine FT = ∆p
2. Find p 4m′ = p 4m(0) + ∆p , and v′
FT = 3mv i
p 4m′ = 3mv i - 4mv j; v′ = 0.75v i - v j
118 ·· An open railroad car of mass 20,000 kg is rolling without friction at 5 m/s along a level track when it starts to rain.
After the car has collected 2000 kg of water, the rain stops. (a) What is the car’s velocity? (b) As the car is rolling
along, the water begins leaking out of a hole in the bottom at a rate of 5 kg/s. What is the velocity after half the water
has leaked out? (c) What is the velocity after all the water has leaked out?
(a) Assume rain falls vertically; use p i = p f
vf = (20,000/22,000) × 5 m/s = 4.55 m/s
(b), (c) Drops fall with v of car, so vcar unchanged
v = 4.55 m/s
119 ·· In the “slingshot effect,” the transfer of energy in an elastic collision is used to boost the energy of a space probe
so that it can escape from the solar system. Figure 8-63 shows a space probe moving at 10.4 km/s (relative to the
sun) toward Saturn, which is moving at 9.6 km/s (relative to the sun) toward the probe. Because of the gravitational
attraction between Saturn and the probe, the probe swings around Saturn and heads back in the opposite direction
with speed vf. (a) Assuming this collision to be a one-dimensional elastic collision with the mass of Saturn much
greater than that of the probe, find vf. (b) By what factor is the kinetic energy of the probe increased? Where does
this energy come from?
(a) 1. For an elastic collision, u app = -u rec
u app = (10.4 + 9.6) km/s = 20 km/s
2. Use Equ. 8-34; vcm = -9.6 km/s
vrec = 29.6 km/s
2 2
(b) Find Kf /Ki = vf /vi
Kf /Ki = (29.6/10.4) 2 = 8.1; energy comes from the slowing
of Saturn (by an immeasurably small amount)
120 ·· You (mass 80 kg) and your friend (mass unknown) are in a rowboat (mass 60 kg) on a calm lake. You are at the
center of the boat rowing and she is at the back, 2 m from the center. You get tired and stop rowing. She offers to
row and after the boat comes to rest, you change places. You notice that after changing places the boat has moved
20 cm relative to a fixed log. What is your friend’s mass?
Take the coordinate origin to be at the end of the boat prior to shifting places.
1. Express xcm prior to shift
xcm = [(60 + 80) × 2/(140 + m)] m
2. Write expression for xcm′, xcm after shift
3. But Fext = 0, so xcm′ = xcm; solve for m
xcm′ = [(60 + m) × (2 ±0.2) ± 80 × 0.2]/(140 + m) m
m = 60 kg or 104 kg
121* ·· A small car of mass 800 kg is parked behind a small truck of mass 1600 kg on a level road (Figure 8-64).
The brakes of both the car and the truck are off so that they are free to roll with negligible friction. A man sitting on
the tailgate of the truck shoves the car away by exerting a constant force on the car with his feet. The car
accelerates at 1.2 m/s2. (a) What is the acceleration of the truck? (b) What is the magnitude of the force exerted on
either the truck or the car?
(a) Fext = 0, ∴ a cm = 0
800 × 1.2 kg.m/s2 = 1600a t; a t = 0.6 m/ss
Chapter 8
Systems of Particles and Conservation of Momentum
(b) F = ma
F = 960 N
122 ·· A 13-kg block is at rest on a level floor. A 400-g glob of putty is thrown at the block such that it travels
horizontally, hits the block, and sticks to it. The block and putty slide 15 cm along the floor. If the coefficient of sliding
friction is 0.4, what is the initial speed of the putty?
1. Find Ki of block + glob; use energy conservation
Ki = f ks = µkMgs = 0.4 × 13.4 × 9.81 × 0.15 J = 7.89 J
2. Find Mvcm after collision; then use p i = p f
Mvcm =
vgl =
2MK i = mgl vgl
2 × 7.89 ×13.4 /0.4 m/s = 36.4 m/s00
123 ·· A careless driver rear-ends a car that is halted at a stop sign. Just before impact, the driver slams on his brakes,
locking the wheels. The driver of the struck car also has his foot solidly on the brake pedal, locking his brakes. The
mass of the struck car is 900 kg, and that of the initially moving vehicle is 1200 kg. On collision, the bumpers of the
two cars mesh. Police determine from the skid marks that after the collision the two cars moved 0.76 m together.
Tests revealed that the coefficient of sliding friction between the tires and pavement was 0.92. The driver of the
moving car claims that he was traveling at less than 15 km/h as he approached the intersection. Is he telling the truth?
vcm = (2 × 0.92 × 9.81 × 0.76) 1/2 m/s = 3.7 m/s
1. Use v = 2 as and a = µsg to find v = vcm
vci = (2100 × 3.7/1200) m/s = 6.475 m/s = 23.3 km/h; No
2. Use p = p to find v , initial speed of moving car
i
f
ci
124 ·· A pendulum consists of a 0.4-kg bob attached to a string of length 1.6 m. A block of mass M rests on a horizontal
frictionless surface (Figure 8-65). The pendulum is released from rest at an angle of 53o with the vertic al and the bob
collides elastically with the block. Following the collision, the maximum angle of the pendulum with the vertical is
5.73o. Determine the mass M.
1. Use energy consevation to find KM
KM = 0.4 × 9.81 × 1.6[cos 5.73o - cos 53o] J = 2.47 J
2. Find vmi and vmf at collision
vmi = [2 × 9.81 × 1.6(1 - cos 53o)]1/2 m/s = 3.544 m/s
vmf = [2 × 9.81 × 1.6(1 - cos 5.73o)]1/2 m/s = 0.396 m/s
3. Use p i = p f to find p M after collision
3.544 × 0.4 = p M ± 0.396 × 0.4;
p M = 1.576 or 1.26 kg.m/s
2
4. Use KM = p M /2M to determine M
M = 0.32 kg or M = 0.50 kg
125* ·· Initially, mass m = 1.0 kg and mass M are both at rest on a frictionless inclined plane (Figure 8-66). Mass M rests
against a spring that has a spring constant of 11,000 N/m. The distance along the plane between m and M is 4.0 m.
Mass m is released, makes an elastic collision with mass M, and rebounds a distance of 2.56 m back up the inclined
plane. Mass M comes to rest momentarily 4.0 cm from its initial position. Find the mass M.
1. Use conservation of energy
mg∆h = 1/2kx2 - Mgx sin 30o; ∆h = 1.44 sin 30o = 0.72 m
2. Find M: x = 0.04 m, m = 1 kg, k = 11×104 N/m
M = kx/g - 2m∆h/x; M = 8.85 kg
126 ·· A circular plate of radius r has a circular hole cut out of it having radius r/2 (Figure 8-67). Find the center of mass
of the plate. Hint: The hole can be represented by two disks superimposed, one of mass m and the other of mass -m.
Chapter 8
Systems of Particles and Conservation of Momentum
By symmetry, xcm = 0. Let σ be the mass per unit area of the disk. Then the mass of the complete disk is π r2σ = M,
and the mass of the material removed is π r2σ/4 = M/4. Thus, ycm = (-M/4)(-r/2)/(3M/4) = r/6.
127 ·· Using the hint from Problem 126, find the center of mass of a solid sphere of radius r that has a spherical cavity of
radius r/2, as in Figure 8-68.
Since V ∝ r3, the mass of the material removed is M/8. Thus, xcm = ycm = 0, and zcm = (-M/8)(-r/2)/(7M/8) = r/14.
128 ·· A neutron of mass m makes an elastic head-on collision with a stationary nucleus of mass M. (a) Show that the
energy of the nucleus after the collision is Knucleus = [4mM/(m + M)2]Kn, where Kn is the initial energy of the neutron.
(b) Show that the fraction of energy lost by the neutron in this collision is
− ∆K n
4mM
4(m / M )
=
.
2 =
Kn
(m + M )
(1 + m / M )2
(a) 1. Write the expression for energy conservation
2. Use momentum conservation
3. Eliminate p nf in (1) using (2); simplify
4. Use (3) to write p ni2/2m in terms of p M
5. Express KM = p M2/2M in terms of Kn
(b) From energy conservation, ∆Kn = -KM
p ni2/2m = p M2/2M + p nf2/2m
p ni = p nf + p M; p nf = p ni - p M
p M/2M + p M/2m - p ni/m = 0
p ni2/2m = Kn = p M2(M + m)2/8M 2m
KM = p M2/2M = Kn[4Mm/(M + m)2]; Q.E.D.
From (5): ∆Kn/Kn = 4Mm/(M + m)2
(1)
(2)
(3)
(4)
(5)
129* ·· The mass of a carbon nucleus is approximately 12 times the mass of a neutron. (a) Use the results of
Problem 128 to show that after N head-on collisions of a neutron with carbon nuclei at rest, the energy of the
neutron is approximately 0.716N E0, where E0 is its original energy. Neutrons emitted in the fission of a uranium
nucleus have an energy of about 2 MeV. For such a neutron to cause the fission of another uranium nucleus in a
reactor, its energy must be reduced to about 0.02 eV. (b) How many head-on collisions are needed to reduce the
energy of a neutron from 2 MeV to 0.02 eV, assuming elastic head-on collisions with stationary carbon nuclei?
(a) 1. Write Knf /Kni = (Kni - ∆Kn)/Kni (see 8-128b)
Knf /Kni = (M - m)2/(M + m)2 = fractional loss per collision
After N collisions, Knf = K0 × 0.716N
2. In this case Knf /Kni = 0.716
-8 = N log(0.716); N = 55
N
-8
(b) In this case (0.716) = 10 ; solve for N
130 ·· On average, a neutron loses 63% of its energy in an elastic collision with a hydrogen atom and 11% of its energy in
an elastic collision with a carbon atom. The numbers are lower than the ones we have been using in earlier problems
because most collisions are not head-on. Calculate the number of collisions, on average, needed to reduce the energy
of a neutron from 2 MeV to 0.02 eV (a desirable outcome for reasons explained in
Problem 129) if the neutron collides with (a) hydrogen atoms and (b) carbon atoms.
(a) In this case, Knf/Kni = 0.37 per collisions.
(see Problem 8-129b) -8 = N log(0.37); N = 19
(b) In this case, Knf/Kni = 0.89 per collision.
-8 = N log(0.89); N = 158
131 ·· A rope of length L and mass M lies coiled on a table. Starting at t = 0, one end of the rope is lifted from the table
with a force F such that it moves with a constant velocity v. (a) Find the height of the center of mass of the rope
as a function of time. (b) Differentiate your result in (a) twice to find the acceleration of the center of mass. (c)
Assuming that the force exerted by the table equals the weight of the rope still there, find the force F you exert on
Chapter 8
Systems of Particles and Conservation of Momentum
the top of the rope.
(a) Let λ = L/M be the mass per unit length. Then ycm = 1/2y(λy)/λL = y2/2L; but y = vt, so ycm = v2t2/2L.
(b) a cm = d 2ycm/dt2 = v2/L.
(c) F = macm + mg, where m = λy = λvt. Thus, F = (M/L)(vt)(v2/L) + (M/L)vtg = Mv3t/L2 + Mvtg/L.
132 ·· A tennis ball of mass mt is held a small distance above a basketball of mass mb. Both are dropped from a height h
above the floor. (Take h to be the distance to the center of the basketball.) The basketball collides elastically with the
floor. Find the speed vt of the tennis ball after it then collides elastically with the basketball. Calculate the height
reached by the tennis ball if mb = 0.480 kg, mt = 0.060 kg, and h = 2 m. (Caution: If you try this experimentally, get
out of the way of the tennis ball!)
The basketball (subscript b) will collide with the floor and rebound first. As it moves up it collides with the tennis ball
(subscript t) which is moving down. We shall first find the velocities of the two balls prior to their collision, and then
use the procedure for elastic collisions to determine the velocity of the tennis ball after the collision.
1. Find vbi, vti, and vcm. Take up as positive.
vbi = 2 gh 0, vti = -vbi; vcm = (mb - mt)vbi/(mb + mt)
2. Find the relative velocity u ti
u ti = vti[1 + (mb - mt)/(mb + mt)] = -u tf
3. Find vtf = u tf + vcm
vtf = 2 gh 0[1 + 2(mb - mt)/(mb + mt)]
4. H = vtf2/2g; use the numerical values given
H = 2[1 + 2(0.48-0.06)/0.54] 2 m = 13.1 m
133* ·· Repeat Problem 24 if the cup has a mass mc and the ball collides with it inelastically.
(a) The same as before, i.e., F = kd + mpg = mg + mpg.
(b) 1. Find vbi; use p i = p f to find vcm of ball + cup
vbi = 2 gh ; vcm = 2 gh [mb/( mc + mb)]0
2. Apply energy conservation
kx2 = (mc+mb)[mb/(mc+mb)]2(2gh) = 2mb2gh/(mb+mc)
3. Solve for the compression x; then multiply by
k to find the force the spring exerts on the
2 gh
2kh
platform.
x=m
; kx = m g
0
b
4. F = mpg + kx
(b) Since the collision is inelastic, the ball never
returns to its original position

k ( mc + mb )
F = g  m p + mb

2kh
g ( mc + mb
b
g (mc + mb )

0


134 ·· Two astronauts at rest face each other in space. One, with mass m1, throws a ball of mass mb to the other, whose
mass is m2. She catches the ball and throws it back to the first astronaut. If they each throw the ball with a speed of v
relative to themselves, how fast are they moving after each has made one throw and one catch?
Note that each collision is perfectly inelastic. Also, vcm = 0. Let vb be velocity of ball.
1. Find v1 and vb after the first throw; use p i = p f
v1 = -mbv/(m1 + mb); vb = m1v/(m1 + mb)
2. Find v2 and vb after m2 catches the ball
v2(m2 + mb) = mbvb; v2 = vmbm1/[(m1 + mb)(m2 + mb)]
3. Find v2 after m2 throws the ball; let this be v2f
v2f = v[mb/(m2 + mb)][1 + m1/(m1 + mb)]
4. Find vbf, velocity of ball after m2 throws it back
vbf = -v{1 - [mb/(m2 + mb)][1 + m1/(m1 + mb)]
Chapter 8
Systems of Particles and Conservation of Momentum
5. Apply momentum conservation to find v1f
6. Simplify the expression for v1f
v1f(m1 + mb) = vbf mb + v1m1
v1f = -m2mbv(2m1 + mb)/[(m1 + mb)2(m2 + mb)]
135 ·· The ratio of the mass of the earth to the mass of the moon is M e/mm = 81.3. The radius of the earth is about 6370
km, and the distance from the earth to the moon is about 384,000 km. (a) Locate the center of mass of the earth–
moon system relative to the surface of the earth. (b) What external forces act on the earth–moon system? (c) In
what direction is the acceleration of the center of mass of this system? (d) Assume that the center of mass of this
system moves in a circular orbit around the sun. How far must the center of the earth move in the radial direction
(toward or away from the sun) during the 14 days between the time the moon is farthest from the sun (full moon) and
the time it is closest to the sun (new moon)?
(a) Use Equ. 8-4; take origin at center of the earth
rcm = mm rem /(M e + mm) = rem /82.3 = 4670 km
(b) Gravitational force of sun (and other planets)
(c) Acceleration is toward the sun
(d) Note that CM is at a fixed distance from sun
d = 2 × 4670 km = 9340 km
136 ·· You wish to enlarge a skating surface so you stand on the ice at one end and aim a hose horizontally to spray
water on the schoolyard pavement. Water leaves the hose at 2.4 kg/s with a speed 30 m/s. If your mass is 75 kg,
what is your recoil acceleration? (Neglect friction and the mass of the hose.)
1. Differentiate p 1 + p 2 = 0
(dm1/dt)v1 + m1a 1 = -(dm2/dt)v2 - m2a 2
2. Here dm1/dt = 2.4 kg/s, a 1 = 0; dm2/dt = 0
a 2 = -(2.4 × 30/75) m/s2 = -0.96 m/s2
137* ·· A neutron at rest decays into a proton plus an electron. The conservation of momentum implies that the
electron and proton should have equal and opposite momentum. However, experimentally they do not. This
apparent nonconservation of momentum led Wolfgang Pauli to suggest in 1931 that there was a third, unseen
particle emitted in the decay. This particle is called a neutrino, and it was finally observed directly in 1957. Suppose
that the electron has momentum p = 4.65 × 10-22 kg.m/s along the negative x direction and the proton
(m = 1.67 × 10-27 kg) moves with speed 2.93 × 105 m/s at an angle 17.9o above the x axis. Find the momentum
of the neutrino. (The kinetic energy of the electron is comparable to its rest energy, so its energy and momentum
are related relativistically rather than classically. However, the rest energy of the proton is large compared with
its kinetic energy so the classical relation E = 1/2mv2 = p 2/2m is valid.)
1. Momentum conservation: p e + p p + p v = 0
p p = 1.67 × 10-27 × 2.93 × 105 kg.m/s
= 4.89 × 10-22 kg.m/s
2. Since p px + p e = 0, p v = -p py j
p px = 4.89 × 10-22cos 17.9o = 4.65 × 10-22 kg.m/s = -p e
-22
o
p v = -4.89 × 10 sin 17.9 j kg.m/s
= -1.5 × 10-22 j kg.m/s
138 ··· A stream of glass beads, each with a mass of 0.5 g, comes out of a horizontal tube at a rate of 100 per second
(Figure 8-69). The beads fall a distance of 0.5 m to a balance pan and bounce back to their original height. How much
mass must be placed in the other pan of the balance to keep the pointer at zero?
1. Find vy of the bead as it hits the pan
vy = (2gh)1/2 = 3.13 m/s
Chapter 8
2. Find ∆p y per bead
3. F = N∆p/∆t = Mg; solve for M
Systems of Particles and Conservation of Momentum
∆p y = 2mvy
M = (100 × 6.26/9.81) N = 63.8 g
139 ··· A dumbbell consisting of two balls of mass m connected by a massless rod of length L rests on a frictionless floor
against a frictionless wall until it begins to slide down the wall as in Figure 8-70. Find the speed v of the bottom ball at
the moment when it equals the speed of the top one.
By symmetry, the speeds will be equal when the angle with the vertical is 45o.
1. Use energy conservation; Ei = Ef
Ei = mgL/2 = Ef = mgL/2 2 0 + 1/2(2m)v2
2. Solve for v
v = [Lg( 2 0 - 1)/2 2 0]1/2 = 1.2 L 0
140 ··· A chain of length L and mass m is held vertically so that the bottom link just touches the floor. It is then dropped.
(a) Show that the acceleration of the top end of the chain is g. (b) If the chain is moving downward with speed v at
time t, and speed v + ∆v at time t + ∆t, find an expression for the change in momentum of the chain during the
interval ∆t. (c) Find the force exerted on the chain by the floor.
(a) Since the chain is in free fall until it hits the floor, the acceleration of each part (other than that which rests
already on the floor) is g.
(b) Let µ = m/L. Then a length dy just above the bottom has a downward speed of gt and momentum given by
dp = µgt dy = µgtv y dt = µg 2t2 dt. Thus, ∆p = µg 2t2∆t
(c) The force exerted on the floor is dp/dt + m′g, where m’ = 1/2µg 2t2 is the mass resting on the floor.
F = (3/2) µg 2t2.