Momentum and Impulse

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Momentum and Impulse
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Objectives

• Calculate the momentum of an object.
• Identify the units of momentum.
• Calculate the momentum of a physical system
  consisting of multiple objects moving in
  different directions.
• Calculate the change in momentum of an object.
• Define impulse and its units.
• Calculate the impulse applied to a physical
  system

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Objectives

• Define the law of conservation of momentum.
• Demonstrate the law of conservation of momentum
  using an interactive simulation.
• Apply the law of conservation of momentum in one
  dimension.

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Physics terms

• momentum
• Impulse
• law of conservation of momentum

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Equations
The momentum of an object is its mass multiplied
by its velocity. Momentum is a vector.
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Equations
Impulse is force multiplied by the time over
which the force acts. The impulse imparted to an
object equals its change in momentum.
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Equations
Conservation of momentum
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What does momentum mean?
How is the word momentum used in everyday
life? Can you think of an example? How is the
physics definition of the word different from the
everyday usage?
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Consider these two objects
A one kilogram sphere is moving at 100 meters per
second. A 100 kilogram sphere is moving at one
meter per second.
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Consider these two objects
If the same stopping force is applied to each,
which sphere will stop first?
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Consider these two objects
If the same stopping force is applied to each,
which sphere will stop first?

1. The 100 kg sphere
2. The 1 kg sphere
3. Its a tie.
4. More information is needed.

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Consider these two objects
If the same stopping force is applied to each,
which sphere will stop first?

1. The 100 kg sphere
2. The 1 kg sphere
3. Its a tie!
4. More information is needed.

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Consider these two objects
If the same stopping force is applied to each,
which sphere will stop first?

1. The 100 kg sphere
2. The 1 kg sphere
3. Its a tie!
4. More information is needed.

Why?
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Momentum
Momentum is the product of mass and velocity.
Momentum was originally identified with a moving
objects persistence of motion.
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Momentum
Momentum is the product of mass and velocity.
p 100 kg m/s
The spheres have the same momentum.
p 100 kg m/s
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Test your knowledge

• A red truck and a blue truck have the same mass.
  The red truck is parked, and the blue truck is
  traveling along the highway at 60 mph.
• Do both trucks have inertia?
• Do both trucks have momentum?

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Test your knowledge

• A red truck and a blue truck have the same mass.
  The red truck is parked, and the blue truck is
  traveling along the highway at 60 mph.
• Do both trucks have inertia?
• Do both trucks have momentum?

Yes. All objects with mass have inertia. They
resist having their motion changed.
18
Test your knowledge

• A red truck and a blue truck have the same mass.
  The red truck is parked, and the blue truck is
  traveling along the highway at 60 mph.
• Do both trucks have inertia?
• Do both trucks have momentum?

Yes. All objects with mass have inertia. They
resist having their motion changed.
No. The blue truck has momentum. The red truck
has NO momentum because it has zero
velocity. Momentum is sometimes referred to as
inertia in motion.
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Units of momentum
Momentum has units of mass multiplied by velocity.
mass in kg
velocity in m/s
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Engaging with the concepts
What is the momentum of a 60 kg sprinter running
at 7.0 m/s?
Momentum
7.0
60
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Engaging with the concepts
What is the momentum of a 60 kg sprinter running
at 7.0 m/s? 420 kg m/s
What is the velocity of the sprinter if her
momentum is 270 kg m/s?
Momentum
7.0
60
420
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Engaging with the concepts
What is the momentum of a 60 kg sprinter running
at 7.0 m/s? 420 kg m/s
What is the velocity of the sprinter if her
momentum is 270 kg m/s? 4.5 m/s

Velocity
270
4.5
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If she wanted to double her momentum, how fast
would she have to run?
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Engaging with the concepts
What is the momentum of a 60 kg sprinter running
at 7.0 m/s? 420 kg m/s
What is the velocity of the sprinter if her
momentum is 270 kg m/s? 4.5 m/s

Velocity
540
9.0
60
If she wanted to double her momentum, how fast
would she have to run? twice as fast (9.0
m/s)
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Engaging with the concepts
A 2,000 kg car and a 4,000 kg truck are both
traveling at 10 m/s when they hit a wall. Which
has more momentum before impact?
What is the ratio of their
momenta?
Momentum
4000
10
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Engaging with the concepts
A 2,000 kg car and a 4,000 kg truck are both
traveling at 10 m/s when they hit a wall. Which
has more momentum before impact? the truck
What is the ratio of
their momenta?
Momentum
40000
4000
10
ptruck pcar is 21
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Engaging with the concepts
A boulder is dropped from rest and hits the
ground at a speed of 15 m/s, transferring
1,200 kg m/s of momentum to the Earth. What is
its mass?
Mass
15
1200
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Engaging with the concepts
A boulder is dropped from rest and hits the
ground at a speed of 15 m/s, transferring
1,200 kg m/s of momentum to the Earth. What is
its mass? 80 kg
Mass
15
1200
80
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Engaging with the concepts
Create two objects with a momentum of 100 kg m/s,
but with masses of 1.0 kg and 4.0 kg.
Velocity
100
1.0
If the mass is four times greater, how does the
velocity change?
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Engaging with the concepts
Create two objects with a momentum of 100 kg m/s,
but with masses of 1.0 kg and 4.0 kg.
Velocity
100
1.0
100
If the mass is four times greater, how does the
velocity change?
The velocity is one-fourth as much.
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Momentum
Momentum is a vector.
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Momentum
Momentum is a vector.
For one-dimensional motion, this means the
direction of motion determines the sign of an
objects momentum.
p -100 kg m/s
p 100 kg m/s
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Momentum of a system
What is the total momentum of this system of two
balls?

• Zero
• 100 kg m/s
• 200 kg m/s

p -100 kg m/s
p 100 kg m/s
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Momentum of a system
What is the total momentum of this system of two
balls?

• Zero! 100 kg m/s -100kg m/s 0 kg m/s
• 100 kg m/s
• 200 kg m/s

p -100 kg m/s
p 100 kg m/s
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Assessment

1. Calculate the momentum of a 1.0 kg object moving
   with a velocity of 20 m/s.
2. What is the velocity of an object that has a
   momentum of -30 kg m/s and a mass of 3.0
   kilograms?
3. Two objects have equal momentum but one has four
   times the mass of the other. What is the
   relationship between their velocities?

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Assessment

1. Calculate the momentum of a 1.0 kg object moving
   with a velocity of 20 m/s.
2. What is the velocity of an object that has a
   momentum of -30 kg m/s and a mass of 3.0
   kilograms?
3. Two objects have equal momentum but one has four
   times the mass of the other. What is the
   relationship between their velocities?

p mv (1.0 kg)(20 m/s) 20 kg m/s
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Assessment

1. Calculate the momentum of a 1.0 kg object moving
   with a velocity of 20 m/s.
2. What is the velocity of an object that has a
   momentum of -30 kg m/s and a mass of 3.0
   kilograms?
3. Two objects have equal momentum but one has four
   times the mass of the other. What is the
   relationship between their velocities?

p mv (1.0 kg)(20 m/s) 20 kg m/s
If p mv, then v p/m (-30 kg m/s)/(3.0 kg)
-10 m/s
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Assessment

1. Calculate the momentum of a 1.0 kg object moving
   with a velocity of 20 m/s.
2. What is the velocity of an object that has a
   momentum of -30 kg m/s and a mass of 3.0
   kilograms?
3. Two objects have equal momentum but one has four
   times the mass of the other. What is the
   relationship between their velocities?

p mv (1.0 kg)(20 m/s) 20 kg m/s
If p mv, then v p/m (-30 kg m/s)/(3.0 kg)
-10 m/s
The lighter object is moving 4 times faster.
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Assessment

• Which answer below shows the correct units for
  momentum?
• kg m/s2
• kg m2/s2
• kg m/s
• kg s/m

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Assessment

• Which answer below shows the correct units for
  momentum?
• kg m/s2
• kg m2/s2
• kg m/s
• kg s/m

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Assessment

• Two bowling balls each have a mass of 4.0 kg.
• The red ball is moving east at 2.0 m/s. The blue
  ball is moving west at 1.0 m/s. Calculate the
  total momentum of the system.

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Assessment

• Two bowling balls each have a mass of 4.0 kg.
• The red ball is moving east at 2.0 m/s. The blue
  ball is moving west at 1.0 m/s. Calculate the
  total momentum of the system.

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Impulse
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Changes in momentum
The momentum of an object changes as it speeds up
or slows down.
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Changes in momentum
The momentum of an object changes as it speeds up
or slows down. For example, this 2000 kg car
slows to a stop and loses its momentum.
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Impulse
A change in momentum is called an impulse,
J. Since impulse is the change in momentum,
it has the same units as momentum kg m/s.
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Impulse
A 1000 kg car is initially parked. It
accelerates to 15 m/s.
a) What is its change in momentum?

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Impulse
A 1000 kg car is initially parked. It
accelerates to 15 m/s.
a) What is its change in momentum? b)
What is the impulse?
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Impulse
A 1000 kg car is initially parked. It
accelerates to 15 m/s.
a) What is its change in momentum? b)
What is the impulse?
the same!
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Calculating impulse
A 500 gram ball of clay is falling at -2.0 m/s
when it strikes the ground. It sticks to the
ground without bouncing. What is the impulse
on the clay during the collision? (Watch out for
signs!)
-2.0 m/s
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Calculating impulse
A 500 gram ball of clay is falling at -2.0 m/s
when it strikes the ground. It sticks to the
ground without bouncing. What is the impulse
on the clay during the collision? (Watch out for
signs!)
-2.0 m/s
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Calculating impulse
2.0 m/s
A 500 gram superball is falling at 2.0 m/s when
it strikes the ground. It bounces back up at 2.0
m/s. Will the impulse on the superball be
greater than or less than the impulse on the
clay?
-2.0 m/s
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Calculating impulse
2.0 m/s
A 500 gram superball is falling at 2.0 m/s when
it strikes the ground. It bounces back up at 2.0
m/s. Will the impulse on the superball be
greater than or less than the impulse on the
clay?
-2.0 m/s
The impulse on the superball will be greater!
It doesnt just come to a stop. It reverses
direction!
Calculate the impulse.
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Calculating impulse
2.0 m/s
A 500 gram superball is falling at 2.0 m/s when
it strikes the ground. It bounces back up at 2.0
m/s. Calculate the impulse
-2.0 m/s
Twice as much impulse as the clay ball!
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Impulse
There are many ways to deliver the same impulse.
Here are three ways to apply an impulse to slow
down a car. In each case the impulse J ?p is
the same. What is different?
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Impulse
The time the impulse is applied is different in
each case.
If the time decreases, then the force must
increase to supply the same impulse.
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Impulse a second definition
A force F exerted for a time ?t applies an
impulse J.
When an impulse J is applied to an object, it
causes a change in momentum ?p.
This definition of impulse leads to a second set
of units for impulse
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Units
This second definition for impulse gives us a
second set of units for impulse newton-seconds,
or N s.
Impulse units N s kg m/s
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Engaging with the concepts
What impulse is imparted to a car if a force of
500 N is applied for 3.0 seconds?
500
Impulse
3.0
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Engaging with the concepts
What impulse is imparted to a car if a force of
500 N is applied for 3.0 seconds?
500
Impulse
1500
3.0
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Engaging with the concepts
If the force on the car is doubled, what happens
to the impulse? If the time that the force is
exerted triples, what happens to the impulse?
1000
Impulse
3.0
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Engaging with the concepts
If the force on the car is doubled, what happens
to the impulse? it doubles If the
time that the force is exerted triples, what
happens to the impulse? it triples
1000
Impulse
3000
3.0
Impulse is directly proportional to force and to
time.
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Impulse is a vector
Impulse J is a
vector. Impulse can be positive or negative
for motion along a line. The direction of the
force determines the direction of the impulse.
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Engaging with the concepts
Jonathan applies his brakes for 0.50 s, which
imparts an impulse of -1000 N s. What force
did the brakes apply to slow down the car?
Force
-1000
0.50
1000
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Engaging with the concepts
Jonathan applies his brakes for 0.50 s, which
imparts an impulse of -1000 N s. What force
did the brakes apply to slow down the car?
-2000 N
-2000
Force
-1000
0.50
1000
A negative force gives a negative impulse. Click
Run to observe the effect on the car.
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Applying what youve learned

• To change an objects momentum, you can apply
• a large force for a short time OR
• a small force for a long time.
• Describe some situations where you want to apply
  a smaller force for a longer time.

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Applying what youve learned

• How do we decrease the force by increasing the
  time in these cases?
• Car collisions
• Sports collisions

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Applying what youve learned

• How do we decrease the force by increasing the
  time in these cases?
• Car collisions
• Sports collisions

bumpers to protect the car, air bags and seat
belts to protect people
helmets and padding to reduce concussions and
broken bones
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Assessment

• Calculate the change in momentum of a 1000 kg car
  that speeds up from 10 m/s to 15 m/s.

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Assessment

• Calculate the change in momentum of a 1000 kg car
  that speeds up from 10 m/s to 15 m/s.

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Assessment

• Which set of units below is NOT correct for
  impulse?

• kg m/s
• N s
• kg m2/s

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Assessment

• Which set of units below is NOT correct for
  impulse?

• kg m/s
• N s
• kg m2/s

Answer C is NOT correct. Answers A and B
are BOTH correct.
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Assessment

1. A 2.0 kg rocket is subjected to a constant force
   of 400 N that accelerates it from rest to a speed
   of 100 m/s.

1. What is the impulse applied to the rocket?

b) How long did this event last?
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Assessment

1. A 2.0 kg rocket is subjected to a constant force
   of 400 N that accelerates it from rest to a speed
   of 100 m/s.

1. What is the impulse applied to the rocket?

b) How long did this event last?
74
Assessment

1. A 2.0 kg rocket is subjected to a constant force
   of 400 N that accelerates it from rest to a speed
   of 100 m/s.

1. What is the impulse applied to the rocket?

b) How long did this event last?
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Advanced
Newtons second law can now be written in a
different way
Substitute in for the acceleration
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Advanced
This form of Newtons second law can even apply
to situations where the mass of an object
changes.
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Conservation of momentum
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Conservation laws
In a closed system, energy is conserved.
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Conservation laws
Consider this closed system containing ...
two frictionless carts with opposing springs.
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Conservation laws
The carts start pinned together
Consider this closed system containing ...
two frictionless carts with opposing springs.
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Conservation laws
When the pin is released, the carts will fly away
from each other. How fast will does each one go?
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Energy conservation
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Energy conservation
The elastic energy in the two springs . . .

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Energy conservation
The elastic energy in the two springs equals the
kinetic energy of both carts after the release.
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Energy conservation
v1
v2
This is one equation (conservation of
energy) with two unknowns v1 and v2.
We cant solve for the final velocities!
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Not enough information
v1
v2
Energy conservation does not tell us whether the
carts move at the same speed or at different
speeds.
87
Not enough information
v1
v2
Energy conservation does not tell us that the
carts move in opposite directions, although we
know that they do.
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A second law is needed!
Suppose the carts have different masses.
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A second law is needed!
Suppose the carts have different masses.
Are the final speeds still the same?
90
A second law is needed!
Energy conservation says nothing about how the
two velocities compare with each other.
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What patterns do you see?
Mass, velocity, momentum, and energy data
92
Equal masses equal speeds
Mass, velocity, momentum, and energy data
93
Twice the mass half the speed
Mass, velocity, momentum, and energy data
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Triple the mass 1/3 the speed
Mass, velocity, momentum, and energy data
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Triple the mass 1/3 the speed
Mass, velocity, momentum, and energy data
What principle is operating here? Notice that the
momentum is equal and opposite!
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Examining the momentum
Mass, velocity, momentum, and energy data
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Conservation of momentum
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Conservation of momentum
The total momentum of a closed system remains
constant.
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Momentum
Momentum is mass times velocity
100
At the start ...
The total momentum is zero.
101
As long as no outside forces act...
The total momentum is zero.
102
As long as no outside forces act...
The total momentum is conserved.
103
One carts momentum is positive .. .
Mass, velocity, momentum, and energy data
104
The other carts momentum is negative
Mass, velocity, momentum, and energy data
105
Total momentum remains zero!
The total momentum remains zero.
106
Momentum is a vector
Momentum is mass times velocity
107
In 1D, direction is given by the sign of the
momentum.
Negative momentum
Positive momentum
108
Why is the law true?
Force Force
By Newtons third law law, the carts put equal
and opposite forces on each other. The TOTAL
force on the system adds to zero.
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Why is the law true?
Force Force
Since the net force on the system is
zero . . . the momentum of the
system cannot change!
110
Apply momentum conservation
A 6.0 kg package explodes into two pieces, A and
B. Piece A, with a mass of 4.0 kg, moves east at
10 m/s.
a) What is the mass of piece B? b) What is
the direction of piece B? c) What is the
speed of piece B?
A
B
10 m/s
4.0 kg
111
Apply momentum conservation
A 6.0 kg package explodes into two pieces, A and
B. Piece A, with a mass of 4.0 kg, moves east at
10 m/s.
a) What is the mass of piece B? 2.0 kg b) What
is the direction of piece B? c) What is the
speed of piece B?
A
B
10 m/s
4.0 kg
112
Apply momentum conservation
A 6.0 kg package explodes into two pieces, A and
B. Piece A, with a mass of 4.0 kg, moves east at
10 m/s.
a) What is the mass of piece B? 2.0 kg b) What
is the direction of piece B? west c) What is
the speed of piece B?
A
B
10 m/s
4.0 kg
113
Apply momentum conservation
A 6.0 kg package explodes into two pieces, A and
B. Piece A, with a mass of 4.0 kg, moves east at
10 m/s.
a) What is the mass of piece B? 2.0 kg b) What
is the direction of piece B? west c) What is
the speed of piece B? Apply conservation of
momentum
A
B
10 m/s
4.0 kg
114
Apply momentum conservation
A 6.0 kg package explodes into two pieces, A and
B. Piece A, with a mass of 4.0 kg, moves east at
10 m/s.

• What is the speed of piece B?

115
Assessment

1. Which statement below correctly summarizes the
   law of conservation of momentum?

• The momentum of an object always remains
  constant.
• The momentum of a closed system always remains
  constant.
• Momentum can be stored in objects such as a
  spring.
• All of the above.

116
Assessment

1. Which statement below correctly summarizes the
   law of conservation of momentum?

• The momentum of an object always remains
  constant.
• The momentum of a closed system always remains
  constant.
• Momentum can be stored in objects such as a
  spring.
• All of the above.

117
Assessment

1. An astronaut with a mass of 100 kg throws a
   wrench with a mass of 2.0 kg at a velocity of 5.0
   m/s. What is the recoil velocity of the
   astronaut if both wrench and astronaut are
   initially at rest?

118
Assessment
This example is physically similar to the
ballistic carts!
119
Assessment

1. An astronaut with a mass of 100 kg throws a
   wrench with a mass of 2.0 kg at a velocity of 5.0
   m/s. What is the recoil velocity of the
   astronaut if both wrench and astronaut are
   initially at rest?

Momentum before Momentum after throwing
wrench throwing wrench
The unknown velocity!
120
Assessment

1. An astronaut with a mass of 100 kg throws a
   wrench with a mass of 2.0 kg at a velocity of 5.0
   m/s. What is the recoil velocity of the
   astronaut if both wrench and astronaut are
   initially at rest?

Momentum before Momentum after throwing
wrench throwing wrench
Initial momentum is zero!
121
Assessment

1. An astronaut with a mass of 100 kg throws a
   wrench with a mass of 2.0 kg at a velocity of 5.0
   m/s. What is the recoil velocity of the
   astronaut if both wrench and astronaut are
   initially at rest?

Momentum before Momentum after throwing
wrench throwing wrench