Download Stacey Carpenter - University of Hawaii System

PPT
Momentum
Developer Notes
Version
04
Date
2004/02/06
Who
dk
05
2004/02/17
dk
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Revisions
Re-formatted
Changed lab activity to keep one cart the same mass
Changed the wording of conservation of momentum
Moved a bunch of the explanation from the teacher
to the student section and eliminated duplication
Re-arranged to put conservation material ahead of
impulse material
Deleted warmup on speed of an aircraft carrier
Goals
 Students should know that momentum is mv.
 Students should know that momentum is conserved in collisions.
 Students should know Ft = ∆(mv).
 Students should know that momentum can be changed by more force in less time or less
force in more time.
Concepts & Skills Introduced
Area
physics
physics
physics
Concept
momentum = mv
impulse and momentum, Ft = ∆(mv)
conservation of momentum
Time Required
Warm-up Question
1. Most people know the word momentum. What do you think momentum is? Describe
momentum.
Presentation
Momentum
Do warm-up question 1. Don't ask for definitions, just what they think it is, related words, etc.
Then lead a discussion, which should follow the pattern below. You can have the students work
to make group descriptions, then discuss it in class. Put the list on the board, then check the
words that come up most often.
Everyone uses the term momentum, but what is it? How is it defined in physics? The word
momentum is often used to mean that something is moving and it will be hard to make it stop or
change directions. What does that sound like to you? Hard to stop or change directions means a
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lot of inertia, or mass. Moving means velocity. So something with a lot of momentum has a lot
of mass and velocity.
Another way to look at it: What do you think of when something is going to hit you? Usually,
how big and how fast. (Also how sharp!) How is mass related to impact, directly or indirectly?
How is velocity related to impact, directly or indirectly? Impact is directly related to mass, and
impact is directly related to velocity, so impact is directly related to massvelocity. That's
momentum.
Do some calculations of momentum just to get a feel for it:
 a 100 kg football player running 5 m/s
 a 10,000 kg truck going 20 m/s.
 a ball bearing at the end of a ramp
 electric car
 What speed would you have to run to have the same momentum as your car going 25 mph?
(mph times 4/9 = m/s)
Conservation of Momentum
Hints for the activity
 The goal of the activity is to discover conservation of momentum, but don't tell the students.
 Make sure the surfaces are flat and level. Make sure the carts roll easily. You can check the
carts by racing them down a ramp. You may need to lube the wheels.
 Make sure the weights are against the rear wall of the carts so that the weights don't slide as
the carts are pushed apart.
Demos/POEs
1. Calculate the mass of a cart + weight from a demo of two carts pushing apart (like the lab),
given the mass of one cart + weight and the distance each cart goes. Could make it tougher
by giving the mass of the second cart.
2. Add mass to a cart that is already rolling. What will happen to its speed? It will slow down the cart had momentum, the brick had none. Together they are in the middle. You can drop a
brick onto a rolling cart - it takes practice and a cushion on the cart.
3. A good example for changing momentum by changing mass instead of velocity is a hot-air
balloon. If the force of the hot air going up and the weight of the balloon are equal, the
balloon will be stationary. Drop some mass off and the velocity will increase.
4. Drop a tennis ball with a playground ball. Drop them together so they are touching like a
snowman, with the tennis ball on top. What will happen after they hit the floor? The tennis
ball rockets off (be careful!) Afterwards, show each ball bouncing on its own, then redemonstrate the two together. The playground ball bounces much lower when they're
together, and the tennis ball much higher.
5. Demonstrate two croquet or billiards balls colliding at various angles.
6. Have the students put several pennies in a row, then shoot another straight into the end. Just
one pops out (like a Newtonian demonstrator, the contraption with 5 suspended ball
bearings).
7. Ball bearing collision off a ramp. If you have the apparatus, run a ball down and off the ramp
and mark where it lands on the floor. Then run the same ball down and collide it with another
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of the same size, marking where they land. Have the students predict how the distances and
angles of the two balls will compare to the single ball. The two balls added together as
vectors should come to the same point as the single ball. This is a nice lab if you have the
equipment.
Impulse & Momentum
See the student section for derivations of impulse and momentum. Later, in the energy unit, we'll
take a look at how long you apply a force using distance instead of time.
There are many, many examples of how long a force is applied affecting the amount of force, or
affecting the velocity. Have the students think of as many as possible.
 To extend the time and lessen the force:
 Bend your knees when landing
 Pad automobile dashboards
 Pad floors for wrestling
 Pad floors for gymnastics
 Use a rubber mallet instead of a steel one
 Use boxing gloves instead of bare fists
 In a car, use springs, shock absorbers, bigger tires, seat cushions,
 Use nylon rope when climbing, or a bungee cord for jumping

 To shorten the time and increase the force:
 Use tighter strings on your tennis racquet
 Use a steel hammer
 Use aluminum baseball bats
 Use plastic bumpers on cars
 Use "crumple zones" on cars
Demos/POEs
1. Like the F=ma lab, run a cart with a weight pulling it, but stop the weight at different times
and look at the speed of the cart.
2. In the same way, use a smaller force for the same amount of time. Or change the mass.
Assessment
Writing Prompts
1.
Relevance
Summary
Answers to Exercises
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1. Which has more momentum, a 1,200 N lineman running at 6 m/s, or an 800 N lineman
running at 9 m/s?
Answer: the same, 720 kgm/s.
2. Two sumo wrestlers start their bout. One of the wrestlers weighs 1,500 N, and the other
weighs 2,000 N. They both lunge forward with the same speed. Which one exerts a larger
force on the other? Which one ends up going backward? They exert equal and opposite
forces. The lighter one goes backward.
3. An 80 kg defensive back runs straight into a 100 kg runner and tackles him. The defensive
back is running 10 m/s and the runner is running 8 m/s. Which one exerts a larger force on
the other? Which one ends up going backward? Answer: The forces are equal and opposite.
Neither goes backward, they stop where they hit.
4. Two identical carts are initially at rest, and a spring between them is released. If cart A has
twice the mass of cart B, what will cart A's velocity be relative to B's?
Answer: Cart a will have 1/2 the velocity of cart B.
5. Auntie and Wendell go to the skating rink. Auntie has a mass of 60 kg and Wendell has a
mass of 20 kg. Wendell is standing still by the side of the rink. Auntie skates by at 2 m/s and
picks him up. What is the speed of Auntie + Wendell after she picks him up?
Answer: Momentum is conserved. Auntie before is 60 kg2 m/s, and Wendell before is 0.
Afterwards, 60 kg2 m/s = 80 kgx m/s. x = 1.5 m/s.
6. A 10 kg fish gets hungry, so he swims up behind a 2 kg fish and swallows it. Before lunch,
the big fish was going 4 m/s, and the little fish was going 2 m/s in the same direction. How
fast is the big fish (+ little fish) going after lunch?
Answer: Momentum is conserved. Momentum before is 10 kg  4 m/s + 2 kg  2 m/s = 44
kgm/s. Momentum after is 12 kg  x m/s = 44 kgm/s. x = 3.67 m/s.
7. Look at the equation Ft = ∆(mv). If force increases, what must happen to keep the equation
balanced? t must go down, or m must go up, or v must go up. The first line of the table is
filled in. Fill in the rest.
IF
F
t 
m
v
THEN
t
F
F
F
OR
m
m
t
t
OR
v
v
v
m
8. Using the equation for momentum, explain why padding in a car helps to protect you during
accidents.
Answer: Your momentum changes during an accident. The padding extends the time so that
the force is less.
9. Using the equation for impulse and momentum, explain why jumpers bend their knees when
they land.
Answer: Your momentum changes to zero when you land. Bending your knees increases the
time so that the force of landing is less.
10. If a car traveling down the highway runs into a bug,
a. which one exerts more force on the other?
Answer: The forces are equal and opposite.
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b. which one hits the other for a longer time?
Answer: The times are the same.
c. which experiences the greater change in momentum?
Answer: The change in momentum must be the same if the force and time are the
same.
d. which experiences the greater change in velocity?
Answer: The bug experiences the greater change in velocity. The car slows down
a little.
e. which experiences the greater acceleration?
Answer: This is the same as d.
11. If a 100 N net force is applied for 10 s to a stationary 5 kg mass, what will its speed be? What
will its momentum be?
Answer: 100 N10 s = 5 kgx m/s. x = 200 m/s. 5 kg 200 m/s = 1,000 kgm/s
Answers to Challenge/ extension
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Background
We've defined momentum as mv. Now play with it a little and see what happens to the
momentum of two objects when they contact each other.
In this lab we use two carts pushed apart by a spring and tied together with a string. We're
looking at their momentum, which is mv. Finding the mass of the carts is easy, but finding the
velocity means having to measure distance and time. Distance is easy to measure, but time is not
as easy to measure accurately. The carts are tied together with a string so that we don't have to
measure time. Here's why: When the carts spring apart, they go different speeds. That means
they go different distances in the same amount of time. The string makes sure they travel for the
same amount of time. Since the time is identical, we can ignore it and just compare distances.
For example if one cart goes 3 m/s, and the other goes 2 m/s, then in one second, the distances
they travel will have the same ratio as their speeds, 3:2.
Problem
Investigate momentum.
Materials
1
mass cart with spring
1
mass cart with no spring
2
bricks? kg masses?
1.5 m string
1
meter stick
1
mass scale
Procedure
 Work in groups of three.
 Your teacher will demonstrate the material setup.
 Set up the equipment in your group.
 Leave one cart at a constant mass. Load the second cart at least four different ways. For each
setup, record each cart's total mass and how far it travels.
Summary
1. We're investigating momentum, which is mv, but you recorded the distances the carts
traveled. How does that compare to their velocities?
Diagram each stage and draw the force vectors affecting the carts' motion.
2. Stage 1 - Describe the initial state of the carts before the spring is released.
3. Stage 2 - Describe what is happening after the spring is released, but while it is still in
contact with both carts.
4. Stage 3 - Describe what happens when the spring is no longer touching both carts, up until
the string reaches its limit.
5. Which cart goes further, the one with more mass or less mass?
6. For each combination, what is the total momentum in stage 1?
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7. For each combination, what is the total momentum in stage 3? (Remember that velocity is a
vector, so direction matters.)
8. Find a pattern in your data, based on the formula for momentum.
9. Try graphing the momentum of cart 1 vs. cart 2.
10. Can you make a general rule that you think might apply in all cases?
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Reading
Everyone uses the term momentum, but what is it? How is it defined in physics? If you were
playing football, and someone was about to tackle you, what are the two main things you'd think
about? Probably how big they are and how fast they're moving.
The word momentum is often used to describe something moving that will be hard to stop or
turn. Does that sound like anything we've studied? How about Newton's 1st Law of Motion? If
something is hard to stop or turn, that means it has a lot of inertia, or mass. And if it is moving,
that means it has velocity.
Momentum is the combination of mass and velocity - momentum equals mass times velocity.
p = mv.
Note that momentum is a vector quantity because velocity is a vector quantity. Direction matters.
Conservation of Momentum
What is momentum good for? It is a measure of the effect something will have. Remember from
Newton's 3rd Law, action-reaction, that an object can only apply as much force as the other
object gives back. In the same way, when two objects hit, they touch for the same amount of
time. So, when two objects hit, they apply the same (but opposite) amount of force to each other
for the same amount of time. F1t1 = -F2t2. That means that ∆m1v1 = -∆m2v2. If two objects
collide, they have equal and opposite changes in momentum. It can't be any other way.
If two objects collide and they always have the same (but opposite) change in momentum, what
does that mean? ∆m1v1 + ∆m2v2 = 0. The total momentum before and after the collision doesn't
change. It is the same! Momentum is conserved! Conservation means that there is the same
amount before and after. Scientists love conservation. It means that they can look at a situation,
and if it violates a law of conservation, they know there is something wrong with their
observations or calculations. Conservation of momentum is one such law.
Conservation of momentum works for more than just two objects. For example, momentum is
conserved in fireworks. When a firework rockets into the air, it has a certain amount of
momentum (mv). When it explodes, little pieces fly off in every direction. If you were able to
find the momentum of every piece and add them together, they would equal the original
momentum before the explosion.
Law of Conservation of Momentum
The total momentum of an isolated system remains constant.
In other words, the sum of momentum before a collision is equal to the sum of momentum after a
collision. ∑mvbefore = ∑mvafter.
Momentum is a new way of combining known quantities, and it gives a new way of looking at
the world, resulting in new knowledge.
Impulse and Momentum
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Remember Newton's 2nd Law of Motion, F = ma? Will it make a difference how long you apply
the force? What does Ft equal? Applying the force for a longer time will result in the same
acceleration, but the acceleration will continue longer, resulting in a greater final speed.
Start with
multiply both sides by t
Remember that
substitute ∆v for at
It can also be written
F = ma
Ft = mat
at = ∆v
Ft = m∆v
Ft = ∆(mv)
This can also be derived from acceleration.
Start with
a = F/m
and
a = ∆v/t
Put them together
F/m = ∆v/t
Rearranged, it's
Ft = m∆v
Same thing as above.
So there's more than one way to achieve the same result. Momentum is defined as mass times
velocity. A change in momentum is achieved by applying a force (surprise!) for a time - that's
called impulse.
Ft = ∆(mv)
Impulse = the change in momentum
Later, in the energy unit, we'll take a look at how long you apply a force using distance instead of
time.
Exercises
1) Which has more momentum, a 1,200 N lineman running at 6 m/s, or an 800 N lineman
running at 9 m/s?
2) Two sumo wrestlers start their bout. One of the wrestlers weighs 1,500 N, and the other
weighs 2,000 N. They both lunge forward with the same speed. Which one exerts a larger
force on the other? Which one ends up going backward?
3) An 80 kg defensive back runs straight into a 100 kg runner and tackles him. The defensive
back is running 10 m/s and the runner is running 8 m/s. Which one exerts a larger force on
the other? Which one ends up going backward?
4) Two identical carts are initially at rest, and a spring between them is released. If cart A has
twice the mass of cart B, what will cart A's velocity be relative to B's?
5) Auntie and Wendell go to the skating rink. Auntie has a mass of 60 kg and Wendell has a
mass of 20 kg. Wendell is standing still by the side of the rink. Auntie skates by at 2 m/s and
picks him up. What is the speed of Auntie + Wendell after she picks him up?
6) A 10 kg fish gets hungry, so he swims up behind a 2 kg fish and swallows it. Before lunch,
the big fish was going 4 m/s, and the little fish was going 2 m/s in the same direction. How
fast is the big fish (+ little fish) going after lunch?
7) Look at the equation Ft = ∆(mv). If force increases, what must happen to keep the equation
balanced? t must go down, or m must go up, or v must go up. The first line of the table is
filled in. Fill in the rest.
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IF
F
t 
m
v
THEN
t
F
F
F
OR
m
m
t
t
OR
v
v
v
m
8) Using the equation for momentum, explain why padding in a car helps to protect you during
accidents.
9) Using the equation for impulse and momentum, explain why jumpers bend their knees when
they land.
10) If a car traveling down the highway runs into a bug,
a) which one exerts more force on the other?
b) which one hits the other for a longer time?
c) which experiences the greater change in momentum?
d) which experiences the greater change in velocity?
e) which experiences the greater acceleration?
11) If a 100 N net force is applied for 10 s to a stationary 5 kg mass, what will its speed be? What
will its momentum be?
Challenge/ extension
1.
Glossary
 Momentum - The mass of an object times its velocity, p = mv.
 Impulse - How long a force is applied, Ft.
 Formula for impulse and momentum - Ft = ∆(mv)
 Law of Conservation of Momentum
The total momentum of an isolated system remains constant.
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