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Physics 4: Linear Momentum and Energy
A.
Momentum and Energy (6-1 to 6-4, 6-10, 7-1 to 7-2)
1. an objects’ momentum (quantity of motion, vector) is
unchanged unless acted upon by a (constant) force for
a period of time (impulse): Ft = mv = p
2. mechanical energy is unchanged unless acted upon
by a (constant) force for a distance (work): W = F||d
a. mechanical energy is energy of motion (thrown
rock) or the potential of moving (rock on a hill)
b. W = Work done on a system
c. measured in joules (1 J = 1 N•m)
d. Only component of F parallel to d does work
F

d
F|| = Fcos  W = (Fcos)d
W>0
W=0
W<0
Fd
Fd
Fd
e. variable force—stretching/compressing a spring
1. graph spring force (Fs) vs. position (x)
Fs = kx
slope = k
Area = W
x
2. Fs = kx  slope = Fs/x = k
3. W = Fsx  area = ½x(kx) = ½kx2 = W
3. kinetic and potential energy
a. intrinsic to object, regardless of direction
b. kinetic energy—energy of motion
1. positive only
2. K = ½mv2 = p2/2m
Steps
Algebra
v2 = v02 + 2ad
start with
assume Ko = 0  vo = 0 v2 = 2ad
ad = ½v2
solve for ad
K = W = Fd
start with
substitute ma for F
K = (ma)d = m(ad)
substitute ½v2 for ad
K = m(½v2)
K = ½mv2
rearrange
mv = p
start with
m2v2 = p2
square both sides
divide both sides by 2m m2v2/2m = p2/2m
substitute K for ½mv2
K = p2/2m
c. potential energy—energy of relative position,
associated with an external force
1. gravitational potential energy
a. based on arbitrary zero
b. Ug = Ug = mgh (near the Earth's surface)
Steps
Algebra
Ug = W = Fd
start with
substitute mg for F
Ug = (mg)d
substitute h for d
Ug = mgh
c. Ug = -GMm/r (orbiting system)
1. G = 6.67 x 10-11 N•m2/kg2
2. r = distance from center to center
3. Ug = 0 when r is Ug < 0 for all
values of r because positive work is
needed reach Ug = 0
2. spring (elastic) potential energy, Us = ½kx2
a. Us = W to stretch or compress the spring
4. power is the rate work is done: P = W/Δt (W)
a. measured in watts (1 W = 1 J/s)
b. P = W/t = F(d/t) = Fvav (v is average)
c. graphing
1. P = W/t  slope of W vs. t graph
2. P = Fvav  area under F vs. v graph
d. kilowatt-hour, 1KWh = 3.6 x 106 J
Name __________________________
B.
Work-Energy Relations (6-5 to 6-9)
1. conservative forces (Fg and Fs)
a. work done does not depend on the path taken
b. work done on object A by a "conservative" force
(gravity, spring) results in the change in form of
mechanical energy (U  K) for object A, but no
change in energy
c. Fg  d : Ug  K, Fg  d : K  Ug
d. Fs  d : Us  K, Fs  d : K  Us
2. non-conservative forces (Ff and Fp)
a. work done depends on the path taken
b. work done on object A by a "nonconservative"
force (push or pull, friction) results in the change
in amount of mechanical energy (ex. friction (W
= Ffd) reduces mechanical energy)
c. mechanical energy is converted into sound,
vibrations of atoms, and temperature increases =
flows as heat—Q
d. total energy is still conserved
3. examples
Type
Process
Energy
pull a pendulum bob off center W  Ug
release pendulum
1
Ug  K
hit a stationary object
KW
throw a ball into the air
WK
K  Ug
2
ball rises and falls
falling ball dents ground
KW
load a projectile in spring-gun W  Us
release projectile
3
Us  K
projectile penetrates target
KW
4. Conservation of energy for work-energy problems

Define boundary of the system or object

Define initial and final states

determine initial energy of the system, Eo
o if elevated h distance: Ug = mgh
o if moving with v velocity: K = ½mv2
o spring stretched/compressed x distance: Us = ½kx2

determine the work done on the system by NC forces
o due to an external push or pull: Wp = F||d
o due to friction: Wf = -(Fn)d = -(mgcosd

d is the distance traveled

 is the angle of incline (0o for horizontal)

determine final energy, Ef
o if height h changes: Ugf = mghf
o If spring stretches or compress x: Usf =½kxf2
o if velocity v changes: Kf = ½mvf2

general equation (not all terms apply for each problem)
K + Ug + Us + W = Kf + Ugf+ Usf
½mv2 + mgh + ½kx2 + F||d = ½mvf2 + mghf + ½kxf2
C. Momentum Conservation and Collisions (7-3 to 7-7)
1. collision between particles doesn't change the total
amount of momentum because the impulse on A equals
the impulse on B, but in the opposite directions
Σpi = Σpf
mAvA + mBvB = mAvAf + mBvBf
Steps
Algebra
FA = -FB
start with Newton's Law
multiply both side by t
FAt = -FBt
substitute mv for Ft
mAvA = -mBvB
mA(vAf – vA) = -mB(vBf – vB)
substitute vf – v for v
-mAvA – mBvB = -mAvAf – mBvBf
collect like v terms
mAvA + mBvB = mAvAf + mBvBf
multiply by -1
1. Perfectly inelastic collisions (vAf = vBf)
a. two particles collide and stick together
b. only momentum is conserved
c. mAvA + mBvB = (mA + mB)vf
Steps
Algebra
mAvA + mBvB = mAvAf + mBvBf
start with
substitute vf for vAf and vBf
mAvA + mBvB = mAvf + mBvf
mAvA + mBvB = (mA + mB)vf
simplify
a. Elastic collisions
1. both energy and momentum are conserved
2. during collision, energy converted to potential
(example: magnetic carts)
a. two unknowns  need two equations
b. mAvA + mBvB = mAvAf + mBvBf
c. ½mAvA2 + ½mBvB2 = ½mAvAf2 + ½mBvBf2
d. simplify to equation: vA + vAf = vB + vBf
mAvA + mBvB =
½mAvA2 + ½mBvB2 =
mAvAf + mBvBf
½mAvAf2 + ½mBvBf2
mA(vA – vAf) = mB(vBf – vB)
mA(vA2 – vAf2) = mB(vBf2 – vB2)
mA(vA2 – vAf2) = mB(vBf2 – vB2)
mA(vA – vAf) = mB(vBf – vB)
(vA – vAf)(vA + vAf) = (vB – vBf)(vB + vBf)
(vA – vAf) = (vBf – vB)
b. Inelastic collisions
2. two particles collide and bounce off
2. only momentum is conserved
3. energy converted to
4. collisions in two dimensions
a. px is conserved independently of py
b. elastic collision
1. mAvAx + mBvBx = mAvAxf + mBvBxf
2. mAvAy + mBvBy = mAvAyf + mBvByf
3. solve two equations & two unknowns
c. perfectly inelastic collision
1. mAvAx + mBvBx = (mA + mB)vxf
2. mAvAy + mBvBy = (mA + mB)vyf
3. solve two equations & two unknowns
d. object explodes into two pieces mA and mB
a.
b.
2.


1. (mA + mB)v = mAvAf + mBvBf
2. opposite inelastic collision equation
solve ballistics problems
M

(vM = 0)
m
vm
bullet collides inelastically with block: mvm = (M + m)vf
block swings or slides (conservation of energy)
o block swings like a pendulum to height h
o K = Ug  ½(M + m)vf2 = (M + m)gh  h = vf2/2g
o block slides a distance d along a rough surface
o K = Wf  ½(M + m)vf2 = (M + m)gd  d = vf2/2g
Practice Problems
A. Momentum and Energy Formulas
Questions 1-2 Two boxes, one heavier than the other, are
initially at rest on a horizontal frictionless surface. The
same constant force F acts on each for 1 s.
1. Which box has more momentum after the force acts?
(A) light
(B) heavy
(C) tie
2. Which box has the greater velocity after the force acts?
(A) light
(B) heavy
(C) tie
3. An open cart rolls along a frictionless track while it is
raining. As it rolls, what happens to the speed of the cart
as the rain collects in it and increases the total mass?
(A) increase
(B) the same
(C) decrease
Questions 4-7 A person pulls a box along a rough, horizontal
floor with force F. Determine if the work done by each force
is positive, zero or negative.
Fn
Ff
F
Fg
4. Fn
(A) W > 0
(B) W = 0
(C) W < 0
5. Fg
(A) W > 0
(B) W = 0
(C) W < 0
6. F
(A) W > 0
(B) W = 0
(C) W < 0
7. Ff
(A) W > 0
(B) W = 0
(C) W < 0
Questions 8-9 Alice applied 10 N of force over 3 m in 10 s. Bill
applies the same force over the same distance in 20 s.
8. Who did more work?
(A) Alice
(B) Bill
(C) tie
9. Who produced the greater power?
(A) Alice
(B) Bill
(C) tie
10. Car A has twice the mass of car B, but they both have the
same kinetic energy. How do their speeds compare?
(A) 2vA = vB
(B) 2vA = vB
(C) 4vA = vB
11. Two balls have total kinetic energy of zero. What can you
say about the total momentum of the two balls?
(A) p > 0
(B) p = 0
(C) p < 0
12. Two balls have total momentum of zero. Does it
necessarily follow that the total kinetic energy is zero?
(A) yes
(B) no
13. Two balls with different masses have the same
momentum. Which ball has greater kinetic energy?
(A) heavier ball
(B) lighter ball
Questions 14-15 Stone A has twice the mass as stone B. They
are dropped from a cliff and reach a point just above ground.
14. What is the speed of stone A compared to stone B?
(A) vA = vB
(B) vA = ½vB
(C) vA = 2vB
(D) vA = 4vB
15. What is the kinetic energy of stone A compared to stone B?
(A) KA = KB
(B) KA = ½KB
(C) KA = 2KB
(D) KA = 4KB
Questions 16-17 Car A and car B are identical, but car A has
twice the velocity of car B.
16. A has kinetic energy K. What is the kinetic energy of B?
(A) 2K
(B) 4K
(C) ½K
(D) ¼K
17. A takes d meters to stop. If B has the same breaking force,
what is B's stopping distance?
(A) 2d
(B) 4d
(C) ½d
(D) ¼d
18. The work Wo accelerates a car for 0 to v. How much work
is needed to accelerate the car from v to 2v?
(A) Wo
(B) 2Wo
(C) 3Wo
(D) 4Wo
19. A golfer making a putt gives the ball an initial velocity of vo,
but the ball only travels one-quarter of the distance to the
hole. What speed should he have given the ball?
(A) 2vo
(B) 3vo
(C) 4vo
(D) 8vo
20. Which can never be negative?
(A) W
(B) K
(C) U
(D) p
21. You and your friend both solve a problem involving a skier
going down a slope, starting from rest. The two of you
have chosen different levels for h = 0 in this problem.
Which of the quantities will you and your friend agree on?
I. Ug
II. Ug
III. K
(A) I only
(B) II only (C) III only (D) II and III
22. Two paths lead to the top of a big hill. A is steep and direct,
while the B is twice as long but less steep. The change in
potential energy on path A compared to path B is
(A) UA < UB
(B) UA = UB
(C) UA > UB
23. How does the work required to stretch a spring 2 cm
compare with the work required to stretch it 1 cm?
(A) W2 = W1
(B) W2 = 2W1
(C) W2 = 4W1
(D) W2 = 8W1
24. A mass attached to a vertical spring moves downward and
stretches the spring. Which is true about the signs for U?
(A) +Us, +Ug
(B) +Us, –Ug
(C) –Us, +Ug
(D) –Us, –Ug
25. What is the momentum of a 0.5-kg ball traveling at 18 m/s?
26. What force is generated by a racket, which strikes a 0.06-kg
tennis ball that reaches a speed of 65 m/s in 0.03 s?
27. How much work is done to move a 50-kg crate horizontally
10 m against a 150-N force of friction?
28. How much work is done to pull a 100-kg crate horizontally
10 m using a force of 100 N at 30o?
29. How much work is done to carry a 100-kg crate 10 m up a
30o ramp?
30. Why is work not needed to keep the Earth orbiting the sun?
31. How much power is needed to change the speed of a
1500-kg car from 10 m/s to 20 m/s in 5 s?
32. How much power does a 75-kg person generate when
climbing 50 steps (rise of 25 cm per step) in 12 s?
a. in Watts
b.
in horse power (1hp = 746 W)
33. How much power is needed to maintain a speed of 25 m/s
against a total friction force of 200 N?
34. How long will it take a 1750-W motor to lift a 285-kg piano
to a sixth-story window 16 m above?
35. A 1000-kg car travels at 30 m/s against 600-N of friction.
a. How much power does the car engine deliver?
b.
The car accelerates at 2 m/s2 from 0 to 30 m/s. What
is the maximum power the car engine must deliver?
c.
The car goes up a 10o incline at 30 m/s. How much
power does the car engine deliver now?
36. What is the kinetic energy of a 2-kg block moving at 9 m/s?
37. What is the gravitational potential energy of a 2-kg block
that is 6 m above zero potential energy?
38. What is the gravitational potential energy of the Earth-Moon
system? (Mearth = 5.97 x 1024 kg, Mmoon = 7.35 x 1022 kg,
distance between Earth and Moon, r = 3.84 x 108 m)
39. Consider the following spring (k = 100 N/m).
a. Calculate the force (F = kx) for each value of x.
x (m)
0.0
0.1
0.2
0.3
0.4
F (N)
b. Graph the data
44. Who makes it to the bottom in the least amount of time?
(A) Alice
(B) Bill
(C) tie
45. A cart starting from rest rolls down a hill and at the bottom
has a speed of 4 m/s. If the cart were given a push, so its
initial speed at the top of the hill was 3 m/s, what would be
its speed at the bottom?
(A) 4 m/s
(B) 5 m/s
(C) 6 m/s
(D) 7 m/s
46. You see a leaf falling to the ground with constant speed.
When you first notice it, the leaf has initial total mechanical
energy Ei. You watch the leaf until just before it hits the
ground, at which point it has final total mechanical energy
Ef. How do these total energies compare?
(A) Ei < Ef
(B) Ei = Ef
(C) Ei > Ef
47. You throw a ball straight up into the air. In addition to
gravity, the ball feels a force due to air resistance.
Compared to the time it takes the ball to go up, the time to
come back down is
(A) less
(B) equal
(C) greater
48. A rock is dropped from 20 m. What is the final velocity?
a. Use kinematics to solve this problem.
b.
Use energy to solve this problem.
0.5
49. A pendulum bob reaches a maximum height of 0.6 m above
the lowest point in the swing, what is its fastest speed?
40 N
50. How far must a 1 kg ball fall in order to compress a spring
0.1 m? (k = 1000 N/m)
20 N
0N
c.
d.
0.2 m
Calculate the area under the graph.
0.4 m
How much potential energy is stored in the stretched
spring?
B. Solving Work-Energy Problems
40. Three balls of equal mass start from rest and roll down
different ramps. All ramps have the same height. Which
ball has the greatest speed at the bottom of the ramp?
 
A
B
C
(A) A
(B) B
(C) C
(D) tie
41. A stationary block slides down a frictionless ramp and
attains a speed of 2 m/s. To achieve a speed of 4 m/s, how
many times higher must the block start from?
(A) 2 times
(B) 4 times
(C) 8 times
42. A box sliding on a frictionless flat surface runs into a fixed
spring, which compresses a distance x to stop the box. If
the initial speed of the box is doubled, how much would the
spring compress?
(A) ½x
(B) x
(C) 2x
(D) 4x
Questions 43-44 Alice and Bill start from rest at the same
elevation on frictionless water slides with different shapes.
 Bill
 Alice
43. At the bottom whose velocity is greater?
(A) Alice
(B) Bill
(C) tie
51. A 10-kg box is initially at the top of a 5-m long ramp set at
53o. The box slides down to the bottom of the ramp. The
force of friction is 31 N. Determine the
a. potential energy at the top of the ramp.
b.
work done by friction during the slide.
c.
velocity of the box at the bottom of the ramp.
52. A spring (k = 500 N/m) is attached to the wall. A 5-kg
block on a horizontal surface ( = 0.25) is pushed against
the spring so that the spring is compressed 0.2 m. The
block is released and propelled across the surface.
a. Determine the potential energy of the spring.
b.
Determine the distance that the block travels.
C. Solving Collision Problems
53. A small car and a large truck collide head-on and stick
together. Which one has larger momentum change?
(A) car
(B) truck
(C) tie
54. A small beanbag and a bouncy rubber ball are dropped
from the same height above the floor. The both have the
same mass. Which would hurt more if it hit you on the
head?
(A) beanbag
(B) rubber ball (C) tie
55. A box slides with initial velocity 10 m/s on a frictionless
surface and collides inelastically with a stationary identical
box. What is the final velocity of the combined boxes?
(A) 0 m/s
(B) 5 m/s
(C) 10 m/s (D) 20 m/s
Questions 56-57 A uranium nucleus (at rest) undergoes fission
and splits into two fragments, one heavy and the other light.
56. Which fragment has the greater momentum?
(A) heavier one (B) lighter one (C) tie
57. Which fragment has the greater speed?
(A) heavier one (B) lighter one (C) tie
58. Alice (50 kg) and Bill (75 kg) are standing on slippery ice
and push off of each other. If Alice slides at 6 m/s, what
speed does Bill have?
(A) 2 m/s
(B) 3 m/s
(C) 4 m/s
(D) 6 m/s
59. A cannon sits on a stationary railroad flatcar with a total
mass of 1000 kg. When a 10-kg cannon ball is fired at a
speed of 50 m/s, what is the recoil speed of the flatcar?
(A) 0 m/s
(B) 0.5 m/s (C) 10 m/s (D) 50 m/s
60. A 25-kg child in a stationary 55-kg boat with a 5-kg
package throws the package out horizontally at 8 m/s.
What is the boat and child's resultant velocity?
vA = + 5 m/s
A = 53o
vB = 3 m/s
B = 90o
66. Stationary mA (1 kg) and mB (3 kg) are separated by a
compressed spring, which is then released. What is the
resulting velocity of mB, when vA is -12 m/s?
67. Determine the velocities after the following elastic
collisions between mA (1 kg) and mB (3 kg).
vA = + 2 m/s
vB = 0 m/s
A = 0o
B = 0o
61. An 85-kg safety running at 5 m/s tackles a 95-kg fullback
traveling at 4 m/s from behind. What is their mutual speed
just after the tackle?
62. A 0.45-kg ice puck, moving east with a speed of 3.0 m/s,
has a head-on elastic collision with a 0.9-kg puck initially at
rest. What are the resulting speeds and directions?
63. A 1-kg block traveling at 5 m/s in the direction of 30o south
of east collides and sticks with a 2-kg block traveling north
at 3 m/s. Determine
a. The x-component of the resulting velocity, vxf.
b.
The y-component of the resulting velocity, vyf.
c.
The resultant speed.
d.
The resultant direction.
64. A 0.50-kg softball is traveling at 40 m/s. A bat makes
contact with the ball for 0.025 s, after which, the ball's
velocity is 35 m/s in the opposite direction (v = -75 m/s).
a. Determine the change in the ball's momentum.
b.
68. A 15-g bullet penetrates a 1.1-kg block of wood. As a
result, the block slides along a surface ( = 0.85) for 9.5 m.
a. How much work is done by friction?
b.
What is the velocity of the system just after the impact?
c.
What is the velocity of the bullet just before the impact?
69. An 18-g bullet traveling at 230 m/s buries itself in a 3.6-kg
pendulum hanging on a 2.8-m long string.
a. Determine for the bullet/pendulum just after impact.
(1) velocity
(2) kinetic energy
b.
70. Andre hits a 0.06-kg tennis ball straight up into the air with
a 300-N force. The ball remains on the racket for 0.25 m.
a. Using dynamics and kinematics, determine
(1) The acceleration.
(2) The initial velocity.
Determine the average force exerted by the bat.
(3) The maximum height reached by the tennis ball.

65. Determine the velocity after the following inelastic
collisions between mA (1 kg) and mB (3 kg).
vA = + 2 m/s
vB = -1 m/s
A = 0o
B = 0o
How high does the pendulum rise?
b.
Use energy to determine the maximum height reached
by the tennis ball.
71. How much power is used to lift 100 kg a distance of 2 m in 4 s?
72. A 1,000-kg car maintains a constant speed of 30 m/s against
a combined friction and air resistance force of 550 N.
a. How much power is needed to cruise at 30 m/s?
b.
2.
A 3-kg block, initially at rest, is
pulled along a frictionless, horizontal surface with a force
shown as a function of time t by the graph.
How steep an incline can the car climb if the engine
can generate 50,000 W of power?
73. A 1-kg block is pushed down against a spring (k = 500 N/m),
which is compressed 0.1 m. The block is released and
propelled vertically.
a. Determine the potential energy of the spring.
b.
Determine the maximum height reached by the block.
The speed of the block at t = 3 s is
(A) 3 m/s
(B) 4 m/s
(C) 6 m/s
3.
74. A 10-kg box is initially at the top of a 5-m long ramp set at
30o. The box slides down to the bottom of the ramp. The
force of friction is 26 N. Determine the
a. potential energy at the top of the ramp.
b.
work done by friction during the slide.
c.
velocity of the box at the bottom of the ramp.
75. A 0.050-kg bullet traveling at 1,000 m/s penetrates a 10-kg
block of wood.
a. What is the velocity of the block after impact?
b.
How far does the block travel along a rough surface
( = 0.25) before stopping?
c.
How high does the block rise if it were suspended
from a long string?
(D) 8 m/s
Two pucks, where mI = 3mII, are attached by a stretched
spring and are initially held at rest on a frictionless surface.
The pucks are released simultaneously. Which is the
same for both pucks as they move toward each other?
(A) Speed
(B) Velocity
(C) Acceleration
(D) Magnitude of momentum
4.
A 2,000-kg railroad car rolls to the right at 10 m/s and
collides and stick to a 3,000-kg car that is rolling to the left at
5 m/s. What is their speed after the collision?
(A) 1 m/s
(B) 2.5 m/s (C) 5 m/s
(D) 7 m/s
5.
A 5-kg block with momentum = 30 kg•m/s, sliding east
across a horizontal, frictionless surface, strikes an obstacle.
The obstacle exerts all impulse of 10 N•s to the west on the
block. The speed of the block after the collision is
(A) 4 m/s
(B) 8 m/s
(C) 10 m/s (D) 20 m/s
6.
In the diagram, a block of mass M initially at rest on a
frictionless horizontal surface is struck by a bullet of mass
m moving with horizontal velocity v.
What is the velocity of the bullet-block system after the
bullet embeds itself in the block?
(A) (M + v)m/M
(B) (m + v)m/M
(C) (m + M)v/M
(D) mv/(m + M)
Practice Multiple Choice
Briefly explain why the answer is correct in the space provided.
1. The force on an object versus time is graphed below.
7.
A disc of mass m is moving to the right with speed v when
it collides and sticks to a second disc of mass 2m. The
second disc was moving to the right with speed v/2.
The object's change in momentum, in kg•m/s, from 0 to 4 s is
(A) 40
(B) 20
(C) 0
(D) -20
The speed of the composite body after the collision is
(A) v/3
(B) v/2
(C) 2v/3
(D) 3v/2
8.
An object of mass m is moving with speed vo to the right on
a horizontal frictionless surface when it explodes into two
pieces. Subsequently, one piece of mass 2/5m moves with
a speed ½vo to the left. The speed of the other piece of
the object is
(A) vo/2
(B) vo/3
(C) 7vo/5
(D) 2 vo
9.
Two objects of mass 0.2 kg and 0.1 kg, respectively, move
parallel to the x-axis. The 0.2 kg object overtakes and
collides with the 0.1 kg object. Immediately after the
collision, the y-component of the velocity of the 0.2 kg
object is 1 m/s upward.
What is the y-component of the velocity of the 0.1 kg object
immediately after the collision?
(A) 2 m/s downward
(B) 0.5 m/s downward
(C) 0 m/s
(D) 0.5 m/s upward
10. Two particles of equal mass mo
moving with equal speeds vo
along paths inclined at 60° to the
x-axis, collide and stick together.
Their velocity after the collision
has magnitude
(A) vo/4
(B) vo/2
(C) vo/2
(D) 3vo/2
11. Student A lifts a 50-N box to a height of 0.4 m in 2.0 s.
Student B lifts a 40-N box to a height of 0.50 m in 1.0 s.
Compared to student A, student B does
(A) the same work but develops more power
(B) the same work but develops less power
(C) more work but develops less power
(D) less work but develops more power
12. What is the change in gravitational potential energy for a
50-kg snowboarder raised a vertical distance of 400 m?
(A) 50 J
(B) 200 J
(C) 20,000 J (D) 200,000 J
19. The graphs show the position d versus time t of three
objects that move along a straight, level path.
Which has no change in kinetic energy?
(A) II only (B) III only (C) I and II (D) I and III
Questions 20-21 A constant force of 900 N pushes a 100 kg
mass up the inclined plane at a uniform speed of 4 m/s.
20. The power developed by the 900-N force is
(A) 400 W (B) 800 W (C) 900 W (D) 3600 W
21. The gain in potential energy when the mass goes from the
bottom of the ramp to the top.
(A) 100 J
(B) 500 J
(C) 1000 J (D) 2000 J
22. What is the maximum height that a 0.1-kg stone rises if 40
J of work is used to shoot it straight up in the air?
(A) 0.4 m (B) 4 m
(C) 40 m
(D) 400 m
23. A 40-N block is released from rest on an incline 8 m above
the horizontal.
13. How high is a 50-N object moved if 250 J of work is done
against the force of gravity?
(A) 2.5 m (B) 10 m
(C) 5 m
(D) 25 m
14. What is the spring potential energy when a spring (k = 80
N/m) is stretched 0.3 m from its equilibrium length?
(A) 3.6 J
(B) 12 J
(C) 7.2 J
(D) 24 J
15. What is the kinetic energy of a 5-kg block that slides down
an incline at 6 m/s?
(A) 20 J
(B) 90 J
(C) 120 J
(D) 240 J
What is the kinetic energy of the block at the bottom of the
incline if 50 J of energy is lost due to friction?
(A) 50 J
(B) 270 J
(C) 320 J
(D) 3100 J
Questions 24-27 The vertical height versus gravitational potential
energy for an object near Earth's surface is graphed below.
Ug (J)
80
Questions 16-17 A weight lifter lifts a mass m at constant speed
to a height h in time t.
16. How much work is done by the weight lifter?
(A) mg
(B) mh
(C) mgh
(D) mght
60
40
20
17. What is the average power output of the weight lifter?
(A) mg
(B) mh
(C) mgh
(D) mgh/t
18. Which is a scalar quantity that is always positive or zero?
(A) Power
(B) Work
(C) Kinetic energy
(D) Potential Energy
h (m)
0.5
1.5
2.5
3.5
24. What is Ug when the object is 2.25 m above the surface?
(A) 50 J
(B) 45 J
(C) 60 J
(D) 55 J
0
25. What is the mass of the object?
(A) 1.5 kg
(B) 2.0 kg
(C) 2.5 kg
(D) 3.0 kg
26. What does the slope of the graph represent?
(A) mass of the object
(B) gravitational force on the object
(C) kinetic energy of the object
(D) potential energy of the object
27. If an object with greater mass was graphed instead of the
object graphed above, how would the slope of the graph
differ from the above graph?
(A) more positive
(B) less positive
(C) equal but negative
(D) be the same
28. A 50-kg diver falls freely from a diving platform that is 3 m
above the surface of the water. What is her kinetic energy
at 1 m above the water?
(A) 0
(B) 500 J
(C) 750 J
(D) 1000 J
29. A 1000 W electric motor lifts a 100 kg safe at constant
velocity. The vertical distance through which the motor
can raise the safe in 10 s is most nearly
(A) 1 m
(B) 3 m
(C) 10 m
(D) 32 m
34. From the top of a 70-m-high building, a 1-kg ball is thrown
directly downward with an initial speed of 10 m/s. If the
ball reaches the ground with a speed of 30 m/s, the energy
lost to friction is most nearly
(A) 50 J
(B) 250 J
(C) 300 J
(D) 450 J

35. A block of mass M is released from rest at the top of an
inclined plane of height h. If the plane is frictionless, what
is the speed of the block at the bottom of the incline?
(A) (2gh)½ (B) 2Mgh
(C) 2MghR2 (D) 5gh

36. Block of mass m slides on a horizontal frictionless table
with an initial speed vo. It then compresses a spring of
force constant k and is brought to rest. How much is the
spring compressed from its natural length?
(A) vo2/2g (B) mgvo/k (C) mvo/k
(D) vo(m/k)½
Practice Free Response
1.
30. Which is the graph of the spring potential energy of a
spring versus elongation from equilibrium?
(A)
(B)
(C)
(D)
A 0.62-kg block is attached to the spring (k = 180 N/m).
When the system is compressed 5.0 cm and released, it
slides a total of 7.3 cm before turning back.
a. How much potential energy does the spring have
when compressed
5.0 cm
when begins
turning back
b. How much work is done by friction as it slides 7.3 cm?
c.
31. Which is the graph of the gravitational potential energy of
an object versus height? (Assume height << Earth's radius)
(A)
(B)
(C)
(D)
32. An object falls freely near Earth's surface. Which graph
best represents the relationship between the object's
kinetic energy and its time of fall?
(A)
(B)
(C)
(D)
33. A system consists of two masses m1 and m2
(m1 < m2). The objects are connected by a string,
hung over a pulley and then released. When the
object of mass m2 has descended a distance h,
the change in potential energy of the system is
(A) (m1 – m2)gh
(B) m2gh
(C) (m1 + m2)gh
(D) ½(m1 + m2)gh
2.
3.
What is the coefficient of friction?
A larger block with mass M slides down a ramp a vertical
distance of 30 cm and strikes a smaller block with mass m,
where M = 3m. The blocks stick and fall 90 cm to the floor.
a. What is the velocity of the larger block just before it
collides with the smaller block?
b.
What is the velocity of the blocks after the collision?
c.
How far horizontally do the blocks travel after colliding?
Students are to calculate the spring constant k of a spring.
The spring is compressed 0.020 m from its uncompressed
length. A mass m is placed on top of the spring. The spring
is then released and the maximum height h reached by the
mass is measured. The students repeat the experiment,
measuring h with various masses m.
a. Derive an expression for the height h in terms of m, x,
k, and fundamental constants.
b.
(1) What quantities should be graphed so that the
slope of a best-fit straight line can be used to
calculate the spring constant k?
(2) Fill in one or both of the blank columns in the
table with calculated values of your quantities.
m (kg)
h (m)
0.020
0.49
0.030
0.34
0.040
0.28
0.050
0.19
0.060
0.18
c.
On the axes below, plot your data and draw a best-fit
straight line. Label the axes and indicate the scale.
d.
Using your best-fit line, calculate the spring constant.