Assignment - Energy and Momentum
Reading Chapters 6 and 7
Objectives/HW
|
|
The student will be able to: |
HW: |
|
1 |
Define
kinetic and potential energy and give examples of each. |
1
– 3 |
|
2 |
Calculate mechanical kinetic energy and
gravitational potential energy (in Joules) and use conservation of energy to
solve related problems. |
4
– 16 |
|
3 |
Define and calculate work. |
17
– 23 |
|
4 |
Relate and equate work and energy and solve
related problems. |
24
– 31 |
|
5 |
Define and calculate power (in |
32
– 39 |
|
6 |
Define and calculate momentum using
appropriate SI units. |
40 |
|
7 |
State and apply the law of conservation of
momentum with proper consideration to internal and external forces. |
41
– 43 |
|
8 |
Use conservation of momentum to solve related
problems. |
44
– 51 |
|
9 |
Define elastic and inelastic collisions and
use the definitions to solve related problems. |
52
– 56 |
Homework Problems
1.
Old-fashioned alarm clocks have springs
that you “wind up”. What type of energy does
the spring have after it is wound? As
the alarm clock runs and performs its various functions the spring loses this
energy – what becomes of it? In other
words, the energy initially in the spring is changed into what other forms?
2.
Describe the energy transformations that
occur when an athlete is pole vaulting.
Trace the changes in energy starting with the athlete standing at rest,
then running, then going up and over the bar, and ending at rest on the big
“cushion” that they land on.
3.
In mountainous areas, road designers
build escape ramps to help stop trucks with failed brakes. These escape ramps are usually roads made of
loose gravel that go up hill. Describe
changes in forms of energy when a fast-moving truck uses one of these escape
ramps.
4.
An earthquake can release energy to
devastate a city. Based on the
conservation principle this energy must come from somewhere. Where does the energy reside right before the
earthquake occurs? And in what form?
5.
As you have already learned, the value of
many physics concepts depends upon and is relative to a frame of
reference. Are values of energy relative
to a frame of reference? Would the change
in an object’s energy be relative to a frame of reference? Explain.
6.
A lump of clay is thrown against the wall
and sticks there. Right before it hits
the lump has both potential energy and kinetic energy. After it hits it still has the same potential
energy but it stops moving. Has energy
been conserved? If so what happened to
the kinetic energy? Explain.
7.
Three rocks are thrown off the top of a
building and into the street below. Each
rock has the same mass and is thrown with the same initial speed. One rock is thrown upward at an angle, one
rock is thrown horizontally, and one is thrown straight down. Make a comparison between the impact
speeds. Which one hits the street with
the greatest speed? The least
speed? Or do they all hit with the same
speed? Explain by referring to energy
concepts.
8.
Mr. M drives his 1300 kg car at speeds around 25 m/s (56 mph).
(a) How much kinetic energy does this represent? (b) Suppose he is passed by a truck with
twice the mass and going 40% faster – 35 m/s (78 mph). How much more kinetic energy does the truck
have?
9.
Throughout time Earth has been pummeled
by impacts from space debris – asteroids, meteoroids, comets, etc. An object the size of
10. Two
floors in a building are separated by 4.1 m.
People move between the two floors on a set of stairs. (a) Determine the change in potential energy
of a 3.0 kg backpack carried up the stairs.
(b) Determine the change in potential energy of a person with weight 650
N that descends the stairs.
11. A
physics student decides to “burn off” some Calories by climbing a ladder. As the ladder is ascended, some food energy
is being converted to gravitational potential energy. (a) If the student has a mass of 75 kg, how
tall a ladder is needed to burn off one Calorie (4190 J)? (b) Conversely, if the student only has a 3.5
m ladder, how much additional mass must he carry up with him to burn off the
Calorie?
12. A
20.0 kg rock falls from the edge of a 100 m cliff. (a) Determine the energy of the rock relative
to the bottom of the cliff. (b)
Determine the impact speed.
13. A
10.0 gram ball is thrown straight down from a height of 2.0 m. The ball strikes the floor at a speed of 7.5
m/s. With what speed was the ball thrown
downward?
14. A
physics book, mass unknown, is dropped 4.50 m.
What will be the impact speed?
15. A
skateboarder with mass 76.5 kg skates back and forth in a "half pipe"
(a semicircle) with radius 8.00 m. (a)
In order to get vertical (i.e. reach the lip of the half pipe), what speed must
the skater have at the lowest point in the pipe? (b) What is the total energy of the
skater? When the skater reaches a point
2.00 m higher than the lowest point find:
(c) the skater's potential energy, (d) the skater's kinetic energy, and
(e) the skater's speed. Use the lowest
point as a reference for PE.
16. A
kid throws a 4.90 N rock off a bridge that is 25.0 m above the water. The initial velocity is 13.0 m/s, 0°. (a) Find the total energy of the rock. (b) Find the rock's speed as it hits the
water. (c) Which results would be
different if the rock was thrown in a different direction? Explain.
17. Describe
any and all situations or circumstances in which a force
acting on an object does: (a) zero work, (b) positive work, and (c) negative
work.
18. A
boy pushes horizontally with 80 N of force on a 20 kg
19. The
third floor of a house is 8.0 m above street level. How much work is needed to move a 152 kg
refrigerator to the third floor?
20. A
sled is pulled across level snow with a force of 225 N along a rope that is
35.0° above
horizontal. If the sled is moved a
distance of 65.3 m, how much work is done?
21. The
escalator at Woodley Park Station is 65 m long and inclined 30°. Determine the work done by the escalator’s
step acting on a 57.0 kg person that rides this distance.
22. A
librarian picks up a 22 N book from the floor to a height of 1.25 m. He then carries the book 8.0 m to the stacks
and places the book on a shelf that is 0.35 m above the floor. Determine how much total work he does on the
book from beginning to end.
23. A
100 kg piano is pushed at a constant speed up a 4.00 m ramp inclined at
an angle of 10.0°. It takes 220 N to push the piano and
there is 50.0 N of friction. (a) Find
the work done by the person pushing. (b)
Find the work done by friction. (c) Find
the work done by gravity.
(d) Find the work done by the normal force.
24. A
ball is dropped from the top of a tall building and reaches terminal velocity
as it falls. Will the potential energy
of the ball upon release equal the kinetic energy it has when striking the
ground? Explain.
25. In
the 1950’s an experimental train that had a mass of 25,000 kg was powered
across a level 500 m track by a jet engine that produced a thrust of 500
kN. (a)
Determine the work done on the train by the engine. (b) Determine the final speed of the train,
ignoring friction.
26. A
2000 kg car with speed 12.0 m/s hits a tree. The tree does not move or break and the front
of the car is smashed inward 50.0 cm.
Ignore friction. (a) Determine
the work done on the car by the tree.
(b) Determine the amount of force involved.
27. A
2200 kg truck encounters an average friction force of 785 N at interstate
speeds. Suppose the truck accelerates
from 25 m/s to 35 m/s over a distance of 350 m.
Determine the amount of force generated by the truck’s drive-train in
order to produce this result.
28. A
222 kg iceboat (like a boat on skates) glides with initially velocity 3.33 m/s
northward across a frozen lake where m=
0.10. The wind then exerts on the boat a
constant force of 444 N, 67.5°
(i.e. NNE) as the boat travels 111 m, 90.0°
(i.e. due north). Find the final speed
of the iceboat.
29. A
constant upward force of 442 N is applied to a stone that weighs 32 N. The upward force is applied through a
distance of 2.0 m, and the stone is then released. To what height, from the point of release,
will the stone rise?
30. An
archer puts a 0.30 kg arrow to the bowstring.
An average force of 201 N is exerted to draw the string back 1.3 m. (a) Assuming no frictional loss, with what
speed does the arrow leave the bow? (b)
If the arrow is shot straight up, how high does it rise?
31. A
girl with mass 28 kg climbs a ladder to the top of a slide, 4.8 m high. She slides down the slide and reaches a speed
of 3.2 m/s at the bottom. Determine the
work done by friction.
32. Two
people of equal mass, Frank and Ernie, climb the same flight of stairs. Frank does it in 25 seconds and Ernie does it
in 35 seconds. (a) Compare the amount of
work – who did more or was it equal?
Explain. (b) Compare the amount
of power – who was more powerful? Or are
they equal? Explain.
33. Brutus,
a champion weightlifter, raises 240 kg a distance of 2.35 m. (a) How much work is done lifting the
weights? (b) How much work is done
holding the weights at rest above his head?
(c) How much work is done lowering them back to the ground? (d) Does Brutus do any work if the weights
are just dropped instead of lowered back to the ground? (e) If the lift is completed in 2.5 s, what
is the power of Brutus?
34. A
certain car’s drive-train produces a force of 5300 N as it accelerates from 0
to 60 mph in 10.0 seconds. If this
acceleration covers a distance of 125 m, determine the power output of the
car’s engine.
35. In
35.0 s, a pump delivers 550 L of oil into a barrel on a platform 25.0 m above
the pump’s intake pipe. The density of
the oil is 0.820 g/cm3. (a) Calculate
the work done by the pump. (b) Calculate
the pump’s power output.
36. A
12.0 m long conveyor belt, inclined at 30.0°,
is used to move bundles of newspapers from the mailroom up to the cargo bay to
be loaded on delivery trucks. Each
newspaper has a mass of 0.78 kg and there are 32 newspapers per bundle. Determine the useful power of the conveyor if
it delivers 15 bundles per minute to the cargo bay.
37. An
engine moves a boat through the water at a constant speed of 15 m/s. The engine must develop a thrust of 6.0 kN to balance the force of drag
from the water acting on the hull.
Determine the power output of the engine.
38. A
188 W motor will lift a load at the rate of 6.50 cm/s. How great a load can this motor lift at this
speed?
39. A
horse walking along the bank tows a barge through a canal. The barge moves due west at 180° through the
canal but the towrope is directed at 200.0°. Tension in the rope is 400 N. (a) How much work is done in pulling the
barge 1.00 km. (b)
If this is a one horsepower horse, how much time is required for the trip?
40. Can
a bullet have the same momentum as a truck?
Explain.
41. If
only an external force can change the momentum of a system or object, how can
the internal force of a car’s brake pads and rotors bring the car to a
stop?
42. NASA
scientists often face a common problem when sending a spacecraft to another
world. The spacecraft will be moving at
tremendous speed to get to the world, but then it must be slowed down in order
to be put into orbit. In light of the
law of conservation of momentum, how is it possible to slow down a spacecraft
in the void and vacuum of space?
43. Two
bullets of equal mass are shot at equal speeds at blocks of wood on a smooth
ice surface. One bullet, made of rubber,
bounces off the wood. The other bullet,
made of aluminum, burrows into the wood.
Which bullet makes the wood move faster?
Why?
44. A
95 kg fullback, running at 8.2 m/s, 0.0°,
collides in midair with a 128 kg defensive tackle moving in the opposite direction. Both players end up with zero speed. (a) What was the fullback’s momentum before
the collision? (b) What was the change
in the fullback’s momentum? (c) What was
the change in the tackle’s momentum? (d)
What was the tackle’s original momentum?
(e) What was the tackle’s speed originally?
45. Ball
A, mass 5.0 g, moves at a velocity of 20.0 cm/s, 180.0°. It collides with Ball B, mass 10.0 g, moving
with velocity 10.0 cm/s, 180.0°. After the collision, ball A is still moving
but with a velocity of 8.0 cm/s, 180.0°. (a) Find the momentum of ball B after the
collision. (b) Find the resulting
velocity of ball B. (c) By how much did
each ball’s momentum change?
46. A
2575 kg van runs into the back of an 825 kg compact car at rest. They move off together at 8.5 m/s. Ignoring friction, find the initial speed of
the van.
47. A
15 g bullet is shot into a 5085 g wooden block standing on a frictionless
surface. The block, with the bullet in
it, acquires a speed of 1.2 m/s.
Calculate the speed of the bullet.
48. A
hockey puck, mass 0.115 kg, moving at 35.0 m/s, strikes an octopus thrown on
the ice by a fan. The octopus has a mass
of 265 g. The puck and octopus slide off
together. Find the speed. (Yes, there truly are hockey fans that throw
octopuses on the ice. Isn’t life
strange?)
49. A
50 kg woman, riding on a 10 kg cart, is moving east at 5.0
m/s. The woman jumps off the cart and
hits the ground running at 7.0 m/s, eastward, relative to the ground. Calculate the velocity of the cart after she
jumps off.
50. A
92 kg fullback, running at 5.0 m/s, attempts to dive across the goal line for a
touchdown. Just as he reaches the goal
line, he is met head-on in midair by two 75 kg linebackers, one moving at 2.0
m/s and the other at 4.0 m/s. If they
all become entangled as one mass, with what velocity do they travel? Does the fullback score?
51. A
10.0 g bullet leaves a rifle with a speed of 800.0 m/s. What should be the minimum mass of the rifle
in order that its recoil speed cannot possibly exceed 1.50 m/s?
52. When
two automobiles collide it will always be an inelastic collision. And if there is friction the total momentum
of the two cars will be reduced in the collision. (a) Is this type of collision a violation of
the law of conservation of energy?
Explain. (b) Is this type of
collision a violation of the law of conservation of momentum? Explain.
53. A
railroad car with mass of 5.0 ´
105 kg collides with a stationary railroad car of equal mass. After the collision, the two cars lock
together and move off at 4.0 m/s. (a)
Determine the initial speed of the first car.
(b) Determine the amount of momentum before and after the
collision. (c) Determine the total
amount of kinetic energy before and after the collision. (d) Explain what becomes of the “missing”
kinetic energy.
54. A
golf ball, mass 0.046 kg, rests on a tee.
It is struck by a golf club with an effective mass of 0.220 kg and a
speed of 44 m/s. Assuming the collision
is perfectly elastic, find the speed of the ball when it leaves the tee.
55. A
steel glider with a mass of 5.00 kg moves along an air track 15.0 m/s, 0° It overtakes and collides with a second
glider of mass 10.0 kg moving in the same direction at 7.50 m/s. After the collision the first glider
continues in the same direction at 7.00 m/s.
(a) With what velocity did the second glider leave the collision? (b) What was the change in momentum of the
first glider? (c) What was the change in
momentum of the second glider? (d) Was
the collision elastic? (prove your answer numerically)
56. A
proton (mass = 1.67 x 10-27 kg) moves with a speed of 6.00
Mm/s. Upon colliding elastically with a
stationary particle of unknown mass, the proton rebounds on its own path with a
speed of 3.60 Mm/s. Find the mass of the
unknown particle.
Answers
– Energy and Momentum
1. After it is wound up the clock has elastic
potential energy (due to the arrangement of the spring). As the clock runs this potential energy is
converted into kinetic energy of the moving hands and internal parts and into
the kinetic energy of the clapper and bells (if the alarm rings). Also, because of friction, some of the
potential energy is converted to heat.
Lastly, any sound that the clock makes represents energy that must have
come from the elastic potential energy initially in the spring.
2. Before starting there is chemical potential
energy in the athlete’s body. As he
runs, this is converted to kinetic energy of the body and the pole. After he “plants” the pole it begins to bend
and energy is “stored” in the pole as elastic potential energy. As he goes up the pole releases its energy
giving the athlete gravitational potential energy. As he falls back down this is converted back
to kinetic energy. Finally, as he lands
energy is converted to heat in the foam pad and sound.
3. The runaway truck has kinetic energy. As it runs up the escape ramp some of this is
converted to gravitational potential energy and some is transferred to the
gravel in the form of kinetic energy – the gravel goes flying.
4. Energy before the earthquake is contained in
the arrangement of the earth’s tectonic plates and is thus a form of potential
energy.
5. The amount of energy does depend on
frame of reference. For example, a tree
has no kinetic energy relative to the surface of the earth because it is not in
motion. However, relative to the sun,
the tree is in motion and does have kinetic energy. The amount of change in energy is also
relative to frame of reference – for example if you accelerate past a tree in a
car, its kinetic energy from the reference frame of the car will appear to
change whereas relative to the earth there is no change.
6. Energy is always conserved – including in
this case when a lump of clay hits a wall and sticks there. By sticking to the wall there is a rapid
decrease in the kinetic energy of the clay (with no corresponding increase in
potential energy). The lost kinetic
energy is changed into sound created by the impact, kinetic energy associated
with vibrations of the wall, and thermal energy associated with a slight increase
in the temperature of the lump of clay and the wall.
7. All three rocks begin with the same kinetic
energy due to identical mass and speed.
The rocks also have the same potential energy because of identical mass
and position. Therefore all three rocks
begin with the same total energy.
Regardless of the direction thrown, once the rock hits the street it
will have lost a certain amount of potential energy and gained a corresponding
amount of kinetic energy so that the total is the same. For this reason all three rocks hit the
street at the same speed with the same kinetic energy and the same potential
energy.
8.
a. 410 kJ
b. 3.9 times more
9.
640 Megatons
(2.7 ´
1018 J)
10.
a. 120 J
b. –2700 J
11.
a. 5.7 m
b. 47 kg
12.
a. 19.6 kJ
b. 44.3 m/s
13.
4.1 m/s
14.
9.39 m/s
15.
a. 12.5 m/s
b. 6000 J
c. 1500 J
d. 4500 J
e. 10.8 m/s
16.
a. 165 J
b. 25.7 m/s
c. If the rock is thrown in a different
direction, neither answer will be different so long as it moves as a projectile
until hitting the water – this because energy and speed are scalar quantities
and do not depend on direction.
(Note: other aspects of the motion would
depend on direction such as velocity, maximum height, time in air, etc.)
17.
a. A force will
do zero work if the object does not move at all or if the object moves in a
direction perpendicular to the force.
b. A force will do positive work if the
object moves at least somewhat in the direction of the force.
c. A force will do negative work if the
object moves at least somewhat in the opposite direction of the force.
18.
a. 800 J
b. –780 J
19.
12 kJ
20.
12.0 kJ
21.
18 kJ
22.
7.7 J
23.
a. 880 J
b. –200 J
c. –681 J
d. 0 J
24.
A ball that reaches terminal velocity while falling through the air is in a
condition where it is not gaining kinetic energy because its speed is constant
and yet it is losing potential energy because its height is decreasing. The potential energy lost by the ball in this
condition must become kinetic energy transferred to the air and thermal energy
associated with a slight increase in the temperature of the ball and air. Therefore a ball dropped from a very tall
building will hit the ground with less kinetic energy than its initial
potential energy.
25.
a. 250 MJ
b. 140 m/s
26.
a. –144 kJ
b. 288 kN
27.
2700 N
28.
14.3 m/s
29.
26 m
30.
a. 42 m/s
b. 89 m
31.
–1200 J
32.
a. Because both people gained the same amount of
potential energy by climbing the stairs the amount of work will be the same.
b. Because Frank does this work in less
time his power is greater – he does work at a greater rate.
33.
a. 5500 J
b. 0 J
c. –5500 J
d. Brutus does no work if he drops the weights
because he exerts zero force in that case.
e. 2200 W
34.
66 kW (89 hp)
35.
a. 110 kJ
b. 3.2 kW
36.
367 W
37.
90 kW
38.
2.89 kN (or 295 kg)
39.
a. 376 kJ
b. 8.40 minutes
40.
A bullet could have as much momentum as
a truck if it is moving at a great enough speed and the truck is moving at a very low speed. Because momentum is mass time velocity, the
ratio of the velocities would have to be the reciprocal of the ratio of the
masses for the momentum to be equal (e.g. if the bullet is one millionth the
mass of the truck it would have to have one million times the speed of the
truck).
41. The internal force of the pads acting on the
rotors does not stop the car – for
example if there are very icy conditions on the road the brakes have little or
no effect. The external friction of the
road surface acting on the tire tread is what stops the car and changes its
momentum. You could say that this
external force is “activated” by the action of the internal force.
42. The spacecraft can fire retrorockets, in
which case a rocket engine fires exhaust gases forward in the direction the
craft is moving. These gases carry away
some of the rocket’s original momentum.
Another possibility is to allow the craft to “swoop” through the
planet’s atmosphere, which transfers some of its momentum to the molecules in
that atmosphere (this is called “aerobraking”).
43. The rubber bullet that bounces off the block
of wood would cause the wood move faster than an aluminum bullet of equal mass
that sticks in the wood. The reason is
that the rubber bullet would undergo a much greater change in momentum; it not
only slows down (like the aluminum bullet), but also comes to a complete stop
and reverses direction. Because the
total momentum of the bullet and block must remain constant, the block will
gain the most momentum from the bullet that undergoes the greatest change in
momentum – the rubber bullet.
44.
a. 780 kg m/s, 0°
b. 780 kg m/s,
180°
c. 780 kg m/s, 0°
d. 780 kg m/s,
180°
e. 6.1 m/s
45.
a. 160 g cm/s, 180°
b. 16 cm/s, 180°
c. 60 g
cm/s
46.
11 m/s
47.
410 m/s
48.
10.6 m/s
49.
5.0 m/s, west
50.
0.041 m/s forward,TD!
51.
5.33 kg
52.
a. Inelastic
means that there is a decrease in kinetic
energy – but that does not mean that there is a decrease in total energy and therefore energy can
still be conserved. The loss in kinetic
energy is accompanied by a corresponding increase in other energy forms such as
heat and sound.
b. The decrease in total momentum can be
attributed to the external force of friction acting on the two cars. This causes a decrease in the momentum “held”
by the cars but this “lost” momentum would be transferred to the roadway and
ultimately to the earth as a whole – in this sense momentum is still conserved.
53.
a. 8.0 m/s
b. 4.0 ´
106 kg m/s
c. 16 MJ; 8.0 MJ
d. The missing 8 million Joules of energy
would become heat and sound.
54.
73 m/s
55.
a. 11.5 m/s, 0°
b. 40.0 kg m/s, 180°
c. 40.0 kg m/s,
0°
d. 844 J before;
784 J after
not elastic
56.
6.67 ´ 10-27
kg