284 Chapter 7 Momentum, Collisions, and the Center of Mass
zero. Therefore ptruck, i = mtruck vtruck, i = (7.50 * 103 kg)vtruck, i =
5.25 * 104 kg # m>s, and vtruck, i = (5.25 * 104 kg # m>s)>(7.50 *
103 kg) = 7.00 m>s.
7-4 (a) inelastic, (b) elastic, (c) inelastic. In (a) the ball collides
with Earth, which is very much more massive than the ball. If
the collision were elastic, the ball would rebound with nearly
the same kinetic energy as it had just before hitting the ground,
and so it would bounce back to its original height. Since this
does not happen, mechanical energy must have been lost and
the collision must have been inelastic. In (b) the total kinetic
energy just after the collision is 125 J + 75 J = 200 J, the same
as just before the collision. So mechanical energy is conserved
and the collision is elastic. In (c) the colliding objects (the dog
and the ball) move together after the collision, so this must be a
completely inelastic collision as described in Section 7-4.
7-5 (d) This is the same situation as Example 7-9, so we can
use the same equation for the final velocity of the ball: vfx =
Fcollision, x Dt>m = (1700 N)(8.6 * 1023 s)>(0.43 kg) = 34 m>s
(about 120 km>h or 76 mi>h).
7-6 (d) If the shell did not explode, its center of mass would
hit the target. The force that blows the cannon shell apart is
internal to the shell, so it does not affect the motion of the
center of mass. So the center of mass still hits the target. The
explosion happens when the shell is halfway along its trajectory, so the half that falls vertically lands 100 m short of the
target (half the horizontal distance from cannon to target).
We saw in Example 7-10 that the center of mass of a system
of two equal masses is halfway between the two masses. So
the other mass must land 100 m on the other side of the
Questions and Problems
In a few problems, you are given more data than you actually
need; in a few other problems, you are required to supply data
from your general knowledge, outside sources, or informed
Interpret as significant all digits in numerical values that have
trailing zeros and no decimal points.
For all problems, use g = 9.80 m>s2 for the free-fall acceleration due to gravity.
• Basic, single-concept problem
•• Intermediate-level problem, may require synthesis of concepts and multiple steps
••• Challenging problem
SSM Solution is in Student Solutions Manual
1. •Using the common definition of the word impulse, comment on physicists’ choice to define impulse as the change in
2. •Starting from Newton’s second law, explain how a collision
that is free from external forces conserves momentum. In other
words, explain how the momentum of the system remains constant with time.
3. •If the mass of a basketball is 18 times that of a tennis ball, can
they ever have the same momentum? Explain your answer. SSM
4. •Two objects have equal kinetic energies. Are the magnitudes
of their momenta equal? Explain your answer.
5. •A glass will break if it falls onto a hardwood floor but not
if it falls from the same height onto a padded, carpeted floor.
Describe the different outcomes in the collision between a glass
and the floor in terms of fundamental physical quantities.
6. •A child stands on one end of a long wooden plank that
rests on a frictionless icy surface. (a) Describe the motion of the
plank when she runs to the other end of the plank. (b) Describe
the motion of the center of mass of the system. (c) How would
your answers change if she had walked the plank rather than
run down it?
7. •Based on what you know about center of mass, why is it
potentially dangerous to step off of a small boat before it comes
to a stop at the dock? SSM
8. •A man and his large dog sit at opposite ends of a rowboat,
floating on a still pond. You notice from shore that the boat
moves as the dog walks toward his owner. Describe the motion
of the boat from your perspective.
9. •Astronomy An asteroid (2007 WD5) passed between Earth
and Mars in 2007. Scientists initially estimated a 4% chance
that the 50-m-wide asteroid would collide with Mars. If the
asteroid had collided with Mars, could it have knocked the planet
out of its orbit? Explain your answer. (For more information on
Near Earth Objects, see neo.jpl.nasa.gov/index.html.)
10. •How would you determine if a collision is elastic or
11. •A recent U.S. patent application describes a “damage
avoidance system” for cell phones, which, upon detecting an
impending, uncontrolled impact with a surface, deploys an airfilled bag around the phone. Explain how this could protect the
cell phone from damage.
12. •Cite two examples of totally inelastic collisions that occur
in your daily life.
13. •Why is conservation of energy alone not
sufficient to explain the
motion of a Newton’s
cradle toy, shown in
Figure 7-24? Consider
the case when two balls
are raised and released
Figure 7-24 Problem 13
14. •After being thrown into the air, a lit firecracker explodes
at the apex of its parabolic flight. (a) Is momentum conserved
before or after the explosion? (b) Is the mechanical energy conserved? (c) What is the path of the center of mass? Neglect the
effects of air resistance and explain your answers.
15. •An arrow shot into a straw target penetrates a distance
that depends on the speed with which it strikes the target. How
does the penetration distance change if the arrow’s speed is
doubled? Be sure to list all the assumptions you make while
arriving at your answer.
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Questions and Problems 285
C. One of the particles continues with the same velocity,
and the other comes to rest.
D. One of the particles continues with the same velocity,
and the other reverses direction at twice the speed.
E. More information is required to determine the final
16. •A large semitrailer truck and a small car have equal momentum. How do their speeds compare?
A. The truck has a much higher speed than the car.
B. The truck has only a slightly higher speed than
C. Both have the same speed.
D. The truck has only a slightly lower speed than
E. The truck has a much lower speed than the car.
17. •A tennis player smashes a ball of mass m horizontally at a
vertical wall. The ball rebounds at the same speed v with which
it struck the wall. Has the momentum of the ball changed, and
if so, what is the magnitude of the change?
E. 4mv SSM
18. •You throw a bouncy rubber ball and a wet lump of clay,
both of mass m, at a wall. Both strike the wall at speed v,
but while the ball bounces off with no loss of speed, the clay
sticks. What is the change in momentum of the clay and ball,
respectively, assuming that toward the wall is the positive
A. 0; mv
B. mv; 0
C. 0; 22mv
D. 2mv; 2mv
E. 2mv; 22mv
19. •Consider a completely inelastic, head-on collision between
two particles that have equal masses and equal speeds. Describe
the velocities of the particles after the collision.
A. The velocities of both particles are zero.
B. Both of their velocities are reversed.
C. One of the particles continues with the same velocity
and the other comes to rest.
D. One of the particles continues with the same
velocity and the other reverses direction at twice
E. More information is required to determine the final
20. •An object is traveling in the positive x direction with speed
v. A second object that has half the mass of the first is traveling
in the opposite direction with the same speed. The two experience a completely inelastic collision. The final x component of
the velocity is
21. •Consider a completely elastic head-on collision between
two particles that have the same mass and the same speed.
What are the velocities after the collision?
A. Both are zero.
B. The magnitudes of the velocities are the same, but the
directions are reversed.
22. •Two small, identical steel balls collide completely elastically. Initially, ball 1 is moving with velocity v1, and ball 2 is
stationary. After the collision, the final velocities of ball 1 and
ball 2 are
A. 12 v1; 12 v1
B. v1; 2v1
C. 0; v1
D. 2v1; 0
E. 2v1; 2v1
23. •Two ice skaters, Lilly and John, face each other while
stationary and push against each other’s hands. John’s mass
is twice that of Lilly. How do their speeds compare after the
A. Lilly’s speed is one-fourth of John’s speed.
B. Lilly’s speed is one-half of John’s speed.
C. Lilly’s speed is the same as John’s speed.
D. Lilly’s speed is twice John’s speed.
E. Lilly’s speed is four times John’s speed.
24. •A friend throws a heavy ball toward you while you
are standing on smooth ice. You can either catch the ball or
deflect it back toward your friend. Which of the following
options will maximize your speed right after your interaction
with the ball?
A. You should catch the ball.
B. You should deflect the ball back toward your friend
at the same speed with which it hit your hand.
C. You should let the ball go past you without
D. It doesn’t matter—your speed is the same regardless
of what you do.
E. You should deflect the ball back toward your friend at
half the speed with which it hit your hand.
25. •Two blocks are released from rest on either side of a frictionless half-pipe (Figure 7-25). Block B is more massive than
block A. The height HB from which block B is released is less
than HA, the height from which block A is released. The blocks
collide elastically on the flat section. After the collision, which
A. Block A rises to a height greater than HA and block B
rises to a height less than HB.
B. Block A rises to a height less than HA and block B
rises to a height greater than HB.
C. Block A rises to height HA and block B rises to
D. Block A rises to height HB and block B rises to
E. The heights to which the blocks rise depends on where
along the flat section they collide. SSM
Figure 7-25 Problem 25
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286 Chapter 7 Momentum, Collisions, and the Center of Mass
26. •Estimate the momentum of a car driving the speed limit
on a freeway.
27. •Sports Estimate the momentum of a fastball thrown by a
major league pitcher.
28. •Sports Estimate the momentum of a tennis ball served by
a professional tennis player.
29. •Medical, Biology Estimate the location of the center of
mass of your body. SSM
30. •Estimate the momentum that a bumblebee has when it
strikes a motorcycle rider.
31. •Astronomy Estimate the momentum of Earth as it orbits
32. •Sports Compare the momentum of a fast-pitch softball to
a major league fastball.
33. •Estimate the impulse delivered to a tennis ball that rebounds
from a practice wall. SSM
34. •A car moving at 20 m>s slams into the back end of a car
stopped at a red light. After the collision, the two cars stick
together. If the cars have the same mass, estimate the distance
the two cars travel before coming to rest. Assume that neither
driver applies his brakes during the collision.
35. •The following gives the force (in newtons) acting on a
2-kg object as a function of time. (a) Make a graph of force
versus time. (b) If the object starts from rest, what is its speed
after 25 s?
7-2 Momentum is a vector that depends on an object’s mass,
speed, and direction of motion
36. •A 10,000-kg train car moves east at 15 m>s. Determine
the momentum of the train car.
37. •Sports The magnitude of the instantaneous momentum of
a 57-g tennis ball is 2.6 kg # m>s. What is its speed? SSM
38. •Determine the initial momentum, final momentum, and
change in momentum of a 1250-kg car initially backing up at
5.00 m>s, then moving forward at 14.0 m>s.
39. •Sports What is the momentum of a 135-kg defensive lineman running at 7.00 m>s?
40. •One ball has four times the mass and twice the speed of
another. (a) How does the momentum of the more massive ball
compare to the momentum of the less massive one? (b) How
does the kinetic energy of the more massive ball compare to the
kinetic energy of the less massive one?
41. •A girl who has a mass of 55.0 kg rides her skateboard to
class at a speed of 6.00 m>s. (a) What is her momentum? (b) If
the momentum of the skateboard itself is 30.0 kg # m>s, what
is its mass? SSM
7-3 The total momentum of a system of objects is conserved
under certain conditions
42. •A 2.00-kg object is moving east at 4.00 m>s when it collides with a 6.00-kg object that is initially at rest. After the
completely elastic collision, the larger object moves east at
1.00 m>s. What is the final velocity of the smaller object after
43. •A 3.00-kg object is moving toward the right at 6.00 m>s.
A 5.00-kg object moves to the left at 4.00 m>s. After the two
objects collide completely elastically, the 3.00-kg object moves
toward the left at 2.00 m>s. What is the final velocity of the
44. •Blythe and Geoff are ice skating together. Blythe has
a mass of 50.0 kg and Geoff has a mass of 80.0 kg. Blythe
pushes Geoff in the chest when both are at rest, causing him
to move away at a speed of 4.00 m>s. (a) Determine Blythe’s
speed after she pushes Geoff. (b) In what direction does she
45. ••An object of mass 3M, moving in the +x direction at speed v0,
breaks into two pieces of mass M
and 2M as shown in Figure 7-26. If
u1 = 45° and u2 = 30°, determine the
final velocities of the resulting pieces
in terms of v0. SSM
v2 = ?
v1 = ?
Figure 7-26 Problem 45
46. •In a game of pool, the cue ball is rolling at 2.00 m>s in a
direction 30.0° north of east when it collides with the eight ball
(initially at rest). The mass of the cue ball is 170 g, but the mass
of the eight ball is only 156 g. After the completely elastic collision, the cue ball heads off at 10.0° north of east and the eight
ball moves off due north. What are the final speeds of each ball
after the collision?
47. •Biology During mating season, male bighorn sheep establish dominance with head-butting contests which can be heard
up to a mile away. When two males butt heads, the “winner” is
the one that knocks the other backward. In one contest a sheep
of a mass 95.0 kg and moving at 10.0 m>s runs directly into a
sheep of mass 80.0 kg moving at 12.0 m>s. Which ram wins the
head-butting contest? SSM
48. ••One way that scientists measure the mass of an unknown
particle is to bounce a known particle, such as a proton or
an electron, off the unknown particle in a bubble chamber.
The initial and rebound velocities of the known particle are
measured from photographs of the bubbles it creates as it
moves; the information is used to determine the mass of the
unknown particle. (a) If a known particle of mass m and initial speed v0 collides elastically, head-on with a stationary unknown particle and then rebounds with speed v, find the mass
of the unknown particle in terms of m, v, and v0. (b) If the
known particle is a proton and the unknown particle is a neutron, what will be the recoil speed of the proton and the final
speed of the neutron?
4/9/13 12:19 PM
Questions and Problems 287
7-4 In an inelastic collision, some of the mechanical
energy is lost
60. •Determine the impulse delivered over the first 10 s to the
object acted on by the force described by the graph (Figure 7-27).
49. •A 10,000-kg train car moving due east at 20.0 m>s collides with and couples to a 20,000-kg train car that is initially
at rest. What is the common velocity of the two-car train after
50. •A large fish has a mass of 25.0 kg and swims at 1.00 m>s
toward and then swallows a smaller fish that is not moving. If
the smaller fish has a mass of 1.00 kg, what is the speed of the
larger fish immediately after it finishes lunch?
51. •A 5.00-kg howler monkey is swinging due east on
a vine. It overtakes and grabs onto a 6.00-kg monkey also
moving east on a second vine. The first monkey is moving at
12.0 m>s at the instant it grabs the second, which is moving at
8.00 m>s. After they join on the same vine, what is their common speed? SSM
52. •A 1200-kg car is moving at 20.0 m>s due north. A 1500-kg
car is moving at 18.0 m>s due east. The two cars simultaneously
approach an icy intersection where, with no brakes or steering,
they collide and stick together. Determine the speed and direction
of the combined two-car wreck immediately after the collision.
1 2 3 4 5 6 7 8 9 10
Figure 7-27 Problem 60
61. •Sports A baseball of mass 0.145 kg is thrown at a speed
of 40.0 m>s. The batter strikes the ball with a force of 25,000
N; the bat and ball are in contact for 0.500 ms. Assuming that
the force is exactly opposite to the original direction of the ball,
determine the final speed of the ball.
62. •A 5.00-kg object is constrained to move along a straight
line. Its initial speed is 12.0 m>s in one direction, and its final
speed is 8.00 m>s in the opposite direction. Complete the graph
of force versus time with appropriate values for both variables
(Figure 7-28). Several answers are correct; just be sure that
your answer is internally consistent.
53. •Sports An 85.0-kg linebacker is running at 8.00 m>s directly
toward the sideline of a football field. He tackles a 75.0-kg
running back moving at 9.00 m>s straight toward the goal
line (perpendicular to the original direction of the linebacker).
Determine their common speed and direction immediately after
7-5 In an elastic collision, both momentum and mechanical
energy are conserved
54. •A 2.00-kg ball is moving at 3.00 m>s toward the right. It
collides elastically with a 4.00-kg ball that is initially at rest.
Determine the velocities of the balls after the collision.
55. •A 10.0-kg block of ice is sliding due east at 8.00 m>s when
it collides elastically with a 6.00-kg block of ice that is sliding
in the same direction at 4.00 m>s. Determine the velocities of
the blocks of ice after the collision. SSM
56. •A 0.170-kg ball is moving at 4.00 m>s toward the right.
It collides elastically with a 0.155-kg ball moving at 2.00 m>s
toward the left. Determine the final velocities of the balls after
57. •A neutron traveling at 2.00 * 105 m>s collides elastically with a deuteron that is initially at rest. Determine the
final speeds of the two particles after the collision. The mass
of a neutron is 1.67 * 10227 kg, and the mass of a deuteron is
3.34 * 10227 kg.
tf = ?
Figure 7-28 Problem 62
63. •Sports A baseball bat strikes a ball when both are moving at 31.3 m>s (relative to the ground) toward each other.
The bat and ball are in contact for 1.20 ms, after which the
ball is traveling at a speed of 42.5 m>s. The mass of the bat
and the ball are 850 g and 145 g, respectively. Calculate the
magnitude and direction of the impulse given to (a) the ball
by the bat and (b) the bat by the ball. (c) What average force
does the bat exert on the ball? (d) Why doesn’t the force shatter the bat?
7-7 The center of mass of a system moves as though all of the
system’s mass were concentrated there
64. •Find the coordinates of the center of mass of the three
objects shown in Figure 7-29 if m1 = 4.00 kg, m2 = 2.00 kg, and
m3 = 3.00 kg. Distances are in meters.
58. •A sudden gust of wind exerts a force of 20.0 N for 1.20 s
on a bird that had been flying at 5.00 m>s. As a result, the bird
ends up moving in the opposite direction at 7.00 m>s. What is
the mass of the bird?
7-6 What happens in a collision is related to the time the
colliding objects are in contact
59. •Determine the average force exerted on your hand as you
catch a 0.200-kg ball moving at 20.0 m>s. Assume the time of
contact is 0.0250 s. SSM
Figure 7-29 Problem 64
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288 Chapter 7 Momentum, Collisions, and the Center of Mass
65. •What are the coordinates of the center of mass for the
combination of the three objects shown in Figure 7-30? The
uniform rod has a mass of10.0 kg, has a length of 30.0 cm,
and is located at x = 50.0 cm. The oval football has a mass of
2.00 kg, a semimajor axis of 15.0 cm,a semiminor axis of 8.00
cm, and is centered at x = 250.0 cm.The spherical volleyball
has a mass of 1.00 kg, a radius of 10.0 cm, and is centered at
y = 230 cm. SSM
Figure 7-30 Problem 65
66. •Four beads each of mass M are attached at various locations to a hoop of mass M and radius R (Figure 7-31). Find the
center of mass of the hoop and beads.
Figure 7-31 Problem 66
67. •Sports A major league baseball has a mass of 0.145 kg.
Neglecting the effects of air resistance, determine the momentum of the ball when it hits the ground if it falls from rest on
the roof of the Metrodome in Minneapolis, Minnesota, a height
of 60.0 m.
68. •Forensic scientists can determine the speed at which a rifle
fires a bullet by shooting into a heavy block hanging by a wire.
As the bullet embeds itself in the block, the block and embedded bullet swing up; the impact speed is determined from the
maximum angle of the swing. (a) Which would make the block
swing higher, a 0.204 Ruger bullet of mass 2.14 g and muzzle
speed 1290 m>s or a 7-mm Remington Magnum bullet of mass
9.71 g and muzzle speed 948 m>s? Assume the bullets enter
the block right after leaving the muzzle of the rifle. (b) Using
your answer in part (a), determine the mass of the block so that
when hit by the bullet it will swing through a 60.0° angle. The
block hangs from wire of length 1.25 m and negligible mass.
(c) What is the speed of an 8.41-g bullet that causes the block
to swing upward through a 30.0° angle?
69. •A friend suggests that if all the people in the United States
dropped down from a 1-m-high table at the same time, Earth
would move in a noticeable way. To test the credibility of
this proposal, (a) determine the momentum imparted to Earth
by 300 million people, of an average mass of 65.0 kg, drop-
ping from 1.00 m above the surface. Assume no one bounces.
(b) What change in Earth’s speed would result? SSM
70. •Sally finds herself stranded on a frozen pond so slippery that she can’t stand up or walk on it. To save herself,
she throws one of her heavy boots horizontally, directly away
from the closest shore. Sally’s mass is 60 kg, the boot’s mass is
5 kg, and Sally throws the boot with speed equal to 30.0 m>s.
(a) What is Sally’s speed immediately after throwing the boot?
(b) Where is the center of mass of the Sally–boot system, relative to where she threw the boot, after 10.0 s? (c) How long
does it take Sally to reach the shore, a distance of 30.0 m away
from where she threw the boot? For all parts, assume the ice
71. •You have been called to testify as an expert witness in a
trial involving a head-on collision. Car A weighs 680 kg and
was traveling eastward. Car B weighs 500 kg and was traveling westward at 72.0 km>h. The cars locked bumpers and slid
eastward with their wheels locked for 6.00 m before stopping.
You have measured the coefficient of kinetic friction between
the tires and the pavement to be 0.750. How fast (in miles per
hour) was car A traveling just before the collision?
72. •A 5000-kg open train car is rolling at a speed of 20.0 m>s
when it begins to rain heavily and 200 kg of water collects
quickly in the car. If only the total mass has changed, what is
the speed of the flooded train car? For simplicity, assume that
all of the water collects at one instant and that the train tracks
73. •An 8000-kg open train car is rolling at a speed of 20.0 m>s
when it begins to rain heavily. After water has collected in the
car, it slows to 19.0 m>s. What mass of water has collected in
the car? For simplicity, assume that all of the water collects at
one instant and that the train tracks are frictionless. SSM
74. •An open rail car of initial mass 10,000 kg is moving at
5.00 m>s when rocks begin to fall into it from a conveyor belt.
The rate at which the mass of rocks increases is 500 kg>s. Find
the speed of the train car after rocks have fallen into the car for
a total of 3.00 s.
75. •A skier and a snowboarder of equal mass collide in an area
where the snow is level. Just before the collision, the skier was
moving south at 8.00 m>s and the snowboarder was moving
west at 12.0 m>s. After the unplanned meeting, the skier slid
45° south of west and the snowboarder slid 45° north of west.
What was the speed of each athlete after the collision, assuming
that the collision was completely elastic?
76. •Sports The sport of curling is quite popular in Canada.
A curler slides a 19.1-kg stone so that it strikes a competitor’s
stationary stone at 6.40 m>s before moving at an angle of 120°
from its initial direction. The competitor’s stone moves off at
5.60 m>s. Determine the final speed of the first stone and the
final direction of the second one.
77. ••Biology Lions can run at speeds up to approximately
80.0 km>h. A hungry 135-kg lion running northward at
top speed attacks and holds onto a 29.0-kg Thomson’s
gazelle running eastward at 60.0 km>h. Find the speed and
direction of travel of the lion–gazelle system just after the lion
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Questions and Problems 289
78. •Biology The mass of a pigeon hawk is twice that of the
pigeons it hunts. Suppose a pigeon is gliding north at a speed
of vP = 23.0 m>s when a hawk swoops down, grabs the pigeon,
and flies off (Figure 7-32). The hawk was flying north at speed
of vH = 35.0 m>s, at an angle u = 45° below the horizontal, at
the instant of the attack. Find the final velocity vector of the
birds just after the attack.
Figure 7-32 Problem 78
79. ••A 12.0-g bullet is fired into a block of wood with speed
v = 250 m>s (Figure 7-33). The block is attached to a spring
that has a spring constant of 200 N>m. The block with the
embedded bullet compresses the spring a distance d = 30.0 cm
to the right, before momentarily coming to a stop. Determine
the mass of the wooden block. SSM
Figure 7-33 Problem 79
80. ••In a ballistic pendulum experiment, a small marble is fired
into a cup attached to the end of a pendulum. If the mass of the
marble is 0.00750 kg and the mass of the pendulum is 0.250 kg,
how high will the pendulum swing if the marble has an initial
speed of 6.00 m>s? Assume that the mass of the pendulum is
concentrated at its end.
81. •••A 2.50-kg object (1) elastically
collides with a 3.60-kg object (2) that is
initially at rest. The less massive object
has a speed of v1 = 4.00 m>s and travels
at an angle of u1 with its original direction after the collision; the more massive object has a speed of v2 = 2.50 m>s
and travels at an angle of u2 after the
collision (Figure 7-34). What are the
initial speed of the less massive object
and the two scattering angles, u1 and u2?
v0 = ?
θ1 = ?
θ2 = ?
Figure 7-34 Problem 81
82. ••A 0.0750-kg ball is thrown at 25.0 m>s toward a brick
wall. (a) Determine the impulse that the wall imparts to the ball
when it hits and rebounds at 25.0 m>s in the opposite direction.
(b) Determine the impulse that the wall imparts to the ball when
it hits and rebounds at an angle of 45.0°. (c) If the ball thrown in
part (b) contacts the wall for 0.0100 s, determine the magnitude
and direction of the force that the wall exerts on the ball.
83. ••The mass of an oxygen nucleus is about 16 times that of
a proton. A proton traveling at 5.00 * 105 m>s collides completely elastically with an oxygen nucleus and leaves the interaction at 5.00 * 105 m>s. (a) What fraction of the initial kinetic
energy is transferred to the oxygen nucleus? (b) Determine the
final directions of the proton and the oxygen nucleus.
4/9/13 12:19 PM