Download The Physics of Renewable Energy

Momentum
and Impulse
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
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.
Physics terms
•
momentum
•
Impulse
•
law of conservation
of momentum
Equations
The momentum of an object is its
mass multiplied by its velocity.
Momentum is a vector.
Equations
Impulse is force multiplied by the
time over which the force acts.
The impulse imparted to an object
equals its change in momentum.
Equations
Conservation of momentum:
Momentum before = Momentum after
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?
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.
Consider these two objects
If the same stopping force
is applied to each, which
sphere will stop first?
Consider these two objects
If the same stopping force
is applied to each, which
sphere will stop first?
A. The 100 kg sphere
B. The 1 kg sphere
C. It’s a tie.
D. More information is
needed.
Consider these two objects
If the same stopping force
is applied to each, which
sphere will stop first?
A. The 100 kg sphere
B. The 1 kg sphere
C. It’s a tie!
D. More information is
needed.
Consider these two objects
If the same stopping force
is applied to each, which
sphere will stop first?
A. The 100 kg sphere
B. The 1 kg sphere
C. It’s a tie!
D. More information is
needed.
Why?
Momentum
Momentum is the product of
mass and velocity.
p = mv
Momentum was originally
identified with a moving object’s
persistence of motion.
Momentum
Momentum is the product of
mass and velocity.
p = mv
The spheres have the same
momentum.
p = 100 kg m/s
p = 100 kg m/s
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.
a) Do both trucks have inertia?
b) Do both trucks have momentum?
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.
a) Do both trucks have inertia?
Yes. All objects with mass have inertia.
They resist having their motion changed.
b) Do both trucks have momentum?
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.
a) Do both trucks have inertia?
Yes. All objects with mass have inertia.
They resist having their motion changed.
b) Do both trucks have momentum?
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”.
Units of momentum
Momentum has units of mass multiplied by velocity.
mass in kg
velocity in m/s
Engaging with the concepts
What is the momentum of
a 60 kg sprinter running at
7.0 m/s?
Momentum
60
7.0
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
420
60
7.0
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
If she wanted to double her
momentum, how fast would
she have to run?
Velocity
270
60
4.5
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
If she wanted to double her
momentum, how fast would
she have to run?
twice as fast (9.0 m/s)
Velocity
540
60
9.0
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?
Momentum
4000
What is the ratio of their
momenta?
10
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
Momentum
40000
What is the ratio of their
momenta?
ptruck: pcar is 2:1
4000
10
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
1200
15
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
1200
80
15
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
If the mass is four times
greater, how does the
velocity change?
1.0
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
If the mass is four times
greater, how does the
velocity change?
The velocity is one-fourth as much.
1.0
100
Momentum
Momentum is a vector.
Momentum
Momentum is a vector.
For one-dimensional motion, this means
the direction of motion determines the
sign of an object’s momentum.
p = -100 kg m/s
p = +100 kg m/s
Momentum of a system
What is the total momentum of this system of two balls?
A. Zero
B. +100 kg m/s
A. +200 kg m/s
p = -100 kg m/s
p = +100 kg m/s
Momentum of a system
What is the total momentum of this system of two balls?
A. Zero!
100 kg m/s + -100kg m/s = 0 kg m/s
A. +100 kg m/s
A. +200 kg m/s
p = -100 kg m/s
p = +100 kg m/s
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?
Assessment
1. Calculate the momentum of a 1.0 kg object moving
with a velocity of +20 m/s.
p = mv = (1.0 kg)(20 m/s) = +20 kg 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?
Assessment
1. Calculate the momentum of a 1.0 kg object moving
with a velocity of +20 m/s.
p = mv = (1.0 kg)(20 m/s) = +20 kg 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?
If p = mv, then v = p/m = (-30 kg m/s)/(3.0 kg) = -10 m/s
3. Two objects have equal momentum but one has four
times the mass of the other. What is the relationship
between their velocities?
Assessment
1. Calculate the momentum of a 1.0 kg object moving
with a velocity of +20 m/s.
p = mv = (1.0 kg)(20 m/s) = +20 kg 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?
If p = mv, then v = p/m = (-30 kg m/s)/(3.0 kg) = -10 m/s
3. Two objects have equal momentum but one has four
times the mass of the other. What is the relationship
between their velocities?
The lighter object is moving 4 times faster.
Assessment
4. Which answer below shows the correct units for momentum?
A.
kg m/s2
B.
kg m2/s2
C.
kg m/s
D.
kg s/m
Assessment
4. Which answer below shows the correct units for momentum?
A.
kg m/s2
B.
kg m2/s2
C.
kg m/s
D.
kg s/m
Assessment
5. 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.
Assessment
5. 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.
Impulse
Changes in momentum
The momentum of an object
changes as it speeds up or
slows down.
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.
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.
Impulse
A 1000 kg car is initially parked. It accelerates to 15 m/s.
a) What is its change in momentum?
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?
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!
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
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
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
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 doesn’t just come to a stop.
It reverses direction!
Calculate the impulse.
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!
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 = Δp is the same. What is different?
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.
Impulse: a second definition
A force F exerted for a time Δt applies an impulse J.
When
an impulse J is applied to an object,
This definition of impulse leads to
it causes
a change
in momentum Δp.
a second
set of
units for impulse:
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
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
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
Engaging with the concepts
If the force on the car is
doubled, what happens to
the impulse?
1000
If the time that the force is
exerted triples, what
happens to the impulse?
Impulse
3.0
Engaging with the concepts
If the force on the car is
doubled, what happens to
the impulse?
it doubles
1000
Impulse
3000
If the time that the force is
exerted triples, what
happens to the impulse?
3.0
it triples
Impulse is directly proportional
to force and to time.
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.
Engaging with the concepts
Jonathan applies his
brakes for 0.50 s, which
imparts an impulse of 1000 N s.
Force
What force did the brakes
apply to slow down the
car?
-1000
0.50
1000
Engaging with the concepts
Jonathan applies his
brakes for 0.50 s, which
imparts an impulse of
-1000 N s.
-2000
What force did the brakes
apply to slow down the
car? -2000 N
Force
-1000
0.50
1000
A negative force gives a negative impulse.
Click [Run] to observe the effect on the car.
Applying what you’ve learned
To change an object’s 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.
Applying what you’ve learned
How do we decrease the force by
increasing the time in these cases?
• Car collisions
• Sports collisions
Applying what you’ve learned
How do we decrease the force by
increasing the time in these cases?
• Car collisions
bumpers to protect the car,
air bags and seat belts to
protect people
• Sports collisions
helmets and padding to
reduce concussions and
broken bones
Assessment
1. Calculate the change in momentum of a 1000 kg car
that speeds up from 10 m/s to 15 m/s.
Assessment
1. Calculate the change in momentum of a 1000 kg car
that speeds up from 10 m/s to 15 m/s.
Assessment
2. Which set of units below is NOT correct for
impulse?
A. kg m/s
B. N s
C. kg m2/s
Assessment
2. Which set of units below is NOT correct for
impulse?
A. kg m/s
B. N s
C. kg m2/s
Answer C is NOT correct.
Answers A and B are BOTH correct.
Assessment
3. 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.
a) What is the impulse applied to the rocket?
b) How long did this event last?
Assessment
3. 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.
a) What is the impulse applied to the rocket?
b) How long did this event last?
Assessment
3. 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.
a) What is the impulse applied to the rocket?
b) How long did this event last?
J = F∆ t
Advanced
Newton’s second law can
now be written in a
different way:
Substitute in for the acceleration:
Advanced
This form of Newton’s second law can even apply to
situations where the mass of an object changes.
Conservation
of momentum
Conservation laws
In a closed system, energy is conserved.
Conservation laws
Consider this closed system containing ...
two frictionless carts with opposing springs.
Conservation laws
The carts start
pinned together
Consider this closed system containing ...
two frictionless carts with opposing springs.
Conservation laws
When the pin is released, the carts will fly away
from each other. How fast will does each one go?
Energy conservation
Energy conservation
The elastic energy in the two springs . . .
Energy conservation
The elastic energy in the two springs equals the
kinetic energy of both carts after the release.
Energy conservation
v1
This is one equation (conservation of energy)
with two unknowns: v1 and v2.
We can’t solve for the final velocities!
v2
Not enough information
v1
v2
Energy conservation does not tell us whether the carts move
at the same speed or at different speeds.
Not enough information
v1
v2
Energy conservation does not tell us that the carts move in
opposite directions, although we know that they do.
A second law is needed!
Suppose the carts have different masses.
A second law is needed!
Suppose the carts have different masses.
Are the final speeds still the same?
A second law is needed!
Energy conservation says nothing about how the
two velocities compare with each other.
What patterns do you see?
Mass, velocity, momentum, and energy data
Equal masses
equal speeds
Mass, velocity, momentum, and energy data
Twice the mass
half the speed
Mass, velocity, momentum, and energy data
Triple the mass
1/3 the speed
Mass, velocity, momentum, and energy data
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!
Examining the momentum
Mass, velocity, momentum, and energy data
Conservation of momentum
Conservation of momentum
The total momentum of a closed system remains constant.
Momentum
Momentum is mass times velocity
At the start ...
The total momentum is zero.
As long as no outside forces act...
The total momentum is zero.
As long as no outside forces act...
The total momentum is conserved.
One cart’s momentum is positive .. .
Mass, velocity, momentum, and energy data
The other cart’s momentum is negative
Mass, velocity, momentum, and energy data
Total momentum remains zero!
The total momentum remains zero.
Momentum is a vector
Momentum is mass times velocity
In 1D, direction is given by
the sign of the momentum.
Negative momentum
Positive momentum
Why is the law true?
Force
Force
By Newton’s third law law, the carts put
equal and opposite forces on each other.
The TOTAL force on the system adds to zero.
Why is the law true?
Force
Force
Since the net force on the system is zero . . .
the momentum of the system cannot change!
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
b) What is the direction of piece B?
c) What is the speed of piece B?
A
4.0 kg
10 m/s
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
b) What is the direction of piece B?
c) What is the speed of piece B?
A
4.0 kg
10 m/s
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
b) What is the direction of piece B? west
c) What is the speed of piece B?
A
4.0 kg
10 m/s
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
b) What is the direction of piece B? west
c) What is the speed of piece B?
A
4.0 kg
Apply conservation of momentum
10 m/s
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.
c) What is the speed of piece B?
Assessment
1. Which statement below correctly summarizes the law of
conservation of momentum?
A. The momentum of an object always remains constant.
B. The momentum of a closed system always remains constant.
C. Momentum can be stored in objects such as a spring.
D. All of the above.
Assessment
1. Which statement below correctly summarizes the law of
conservation of momentum?
A. The momentum of an object always remains constant.
B. The momentum of a closed system always remains constant.
C. Momentum can be stored in objects such as a spring.
D. All of the above.
Assessment
2. 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?
Assessment
This example is physically similar
to the ballistic carts!
Assessment
2. 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
throwing wrench
=
Momentum after
throwing wrench
The unknown
velocity!
Assessment
2. 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
throwing wrench
=
Initial momentum is zero!
Momentum after
throwing wrench
Assessment
2. 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
throwing wrench
=
Momentum after
throwing wrench