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Chapter 6
Momentum and Collisions
Momentum and Impulse
Objectives:
1. Compare the momentum of different moving
objects.
2. Compare the momentum of the same object
moving with different velocities.
3. Identify some examples of change in the
momentum of an object.
4. Describe changes in momentum in terms of force
and time.
Linear Momentum
Momentum (p) is a quantity defined as the product
of the mass and velocity of an object.
p = mv
momentum = mass x velocity
SI units of kg▪m/s
The faster an object moves, the more momentum it
has and the more difficult it is to stop the object.
Concept Check
A deer with a mass of 146 kg is running head-on
towards you with a speed of 17 m/s. You are
going north. Find the momentum of the deer.
Concept Check
2.5 x 103 kg▪m/s south
Concept Check
A 21 kg child on a 5.9 kg bike is riding with a
velocity of 4.5 m/s to the northwest.
a. What is the total momentum of the child and the
bike together?
b. What is the momentum of the child?
c. What is the momentum of the bike?
Concept Check
a. 1.2 x 102 kg▪m/s northwest
b. 94 kg▪m/s northwest
c. 27 kg▪m/s northwest
Concept Check
What velocity must a 1210 kg car have in order to
have a momentum of 5.6 x 104 kg▪m/s east?
Concept Check
46 m/s east.
Changes in Momentum
A change in momentum requires force and time.
It requires more force to stop an object with a
greater momentum than to stop an object with a
smaller momentum.
When Newton first expressed his second law, he
wrote:
F = ∆p
∆t
force = change in momentum
time interval
Impulse-Momentum Theorem
We can rearrange Newton's second law such that
F∆t = ∆p or F∆t = ∆p = mvf – mvi
force x time interval = change in momentum
Thus, a small force acting over a long time can
cause the same change in momentum as a large
force acting over a short time.
Assume all forces are constant unless stated
otherwise.
Changes in Momentum
●
The expression F∆t is the impulse of the force over the time
interval.
– The product of the force and the time over
which the force
acts on an object.
Concept Check
A 0.50 kg football is thrown with a velocity of 15
m/s to the right. A stationary receiver catches the
ball and brings it to rest in 0.020 s. What is the
force exerted on the ball by the receiver?
Concept Check
3.8 x 102 N to the left
Concept Check
An 82 kg man drops from rest on a diving board 3.0
m above the surface of the water and comes to rest
0.55 s after reaching the water. What is the net
force on the diver as he is brought to rest?
Concept Check
1.1 x 103 N upward
Concept Check
A 0.40 kg soccer ball approaches a player
horizontally with a velocity of 18 m/s to the north.
The player strikes the ball and causes it to move in
the opposite direction with a velocity of 22 m/s.
What impulse was delivered to the ball by the
player?
Concept Check
16 kg▪m/s to the south
Concept Check
A 0.50 kg object is at rest. A 3.00 N force to the
right acts on the object during a time interval of
1.50 s.
a. What is the velocity of the object at the end of
this time interval?
b. At the end of this interval, a constant force of
4.00 N to the left is applied for 3.00 s. What is the
velocity at the end of the 3.00 s?
Concept Check
a. 9.0 m/s to the right.
b. 15 m/s to the left.
Stopping Times and Distances
Stopping distances and safe following distances are
determined using the impulse-momentum theorem.
Stopping time is determined by solving the impulsemomentum theorem for time.
∆t = ∆ p = mvf-mvi
F
F
Stopping distance can then be determined by the
kinematic equation
∆x = ½ (vi + vf)∆t
Concept Check
If the maximum coefficient of kinetic friction
between a 2300 kg car and a road is 0.50, what is
the minimum stopping distance for a car entering a
skid at 29 m/s?
Concept Check
86 m
Concept Check
How long would it take a 2240 kg car to come to a
full stop from 20.0 m/s to the west if the force on
the car 8410 N to the east? How far would the car
go before stopping? Assume a constant
acceleration.
Concept Check
5.33 s; 53.3 m to the west
Concept Check
A 2500 kg car traveling to the north is slowed down
uniformly from an initial velocity of 20.0 m/s by a
6250 N braking force acting opposite the car's
motion. Use the impulse-momentum theorem to
answer the following questions:
a. What is the car's velocity after 2.50 s?
b. How far does the car move during 2.50 s?
c. How long does it take the car to come to a
complete stop?
Concept Check
a. 14 m/s north
b. 42 m north
c. 8.0 s
Concept Check
Assume the car in problem 2 has a mass of 3250 kg.
a. How much force would be required to cause the
same acceleration as in item 1? Use the impulsemomentum theorem.
b. How far would the car move before stopping?
Concept Check
a. 1.22 x 104 N to the east
b. 53.3 m to the west
More Time = Less Force
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The impulse-momentum theorem is also used to design
safety equipment that reduces the force acting on the
human body during collisions. (Nets, air mattresses, air
bags, etc.)
When an egg is dropped on the floor, it breaks.
When an egg is dropped on a pillow, it doesn't break.
Conservation of Momentum
Objectives:
1. Describe the interaction between two objects in
terms of the change in momentum of each object.
2. Compare the total momentum of two objects
before and after they interact.
3. State the law of conservation of momentum.
4. Predict the final velocities of objects after
collisions, given the initial velocities.
Conservation of Momentum
Momentum is a conserved quantity.
When 2 objects collide, the initial momentum is
equal in magnitude to the final momentum.
m1v1,i + m2v2,i = m1v1,f + m2v2,f
For an isolated system, the law of the conservation
of momentum can be stated as
The total momentum of all objects interacting
with one another remains constant regardless
of the nature of the forces between the objects.
Collisions
●
●
Momentum is conserved in collisions.
Consider two billiard balls: one is moving at a constant
velocity and the other is at rest.
– When the moving ball collides with the nonmoving ball,
the moving ball slows down and the ball that was initially
at rest begins to move.
– The individual momentums of the balls have changed,
but the total momentum of the system remains the
same (IN
THE ABSENCE OF
FRICTION!!!).
● In most problems, disregard friction
Moving Apart
●
●
●
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Conservation of momentum also occurs when objects move
away from each other.
When you are sitting in your favorite class (Physics of
course), your momentum is zero.
When the bell rings and you jump up and run like heck to
the door, you have momentum, so how is momentum
conserved?
You have to include the Earth in your analysis of the
situation.
If you jump up at an initial momentum of 25 kg▪m/s upward
Moving Apart
●
The Earth is moving away from you at a momentum of 25
kg▪m/s downward, so the total momentum remains the
same (zero).
Concept Check
A 63.0 kg astronaut is on a spacewalk when the
tether line to the shuttle breaks. The astronaut is
able to throw a spare 10.0 kg oxygen tank in a
direction away from the shuttle with a speed of
12.0 m/s, propelling the astronaut back to the
shuttle. Assuming that the astronaut starts from
rest with respect to the shuttle, find the astronaut's
final speed with respect to the shuttle after the tank
is thrown.
Concept Check
1.90 m/s
Concept Check
An 85.0 kg fisherman jumps from a dock into a
135.0 kg rowboat at rest on the west side of the
dock. If the velocity of the fisherman is 4.30 m/s
to the west as he leaves the dock, what is the final
velocity of the fisherman and the boat?
Concept Check
1.66 m/s west
Concept Check
Each croquet ball in a set has a mass of 0.50 kg.
The green ball, traveling at 12.0 m/s, strikes the
blue ball, which is at rest. Assuming that the balls
slide on a frictionless surface and all collisions are
head-on, find the final speed of the blue ball in
each of the following situations:
a. The green ball stops moving after it strikes the
blue ball.
b. The green ball continues moving after the
collision at 2.4 m/s in the same direction.
Concept Check
a. 12.0 m/s
b. 9.6 m/s
Concept Check
A boy on a 2.0 kg skateboard initially at rest tosses
a 8.0 kg jug of water in the forward direction. If
the jug has a speed of 3.0 m/s relative to the
ground and the boy and the skateboard move in the
opposite direction at 0.60 m/s, find the boy's mass.
Concept Check
38 kg
Forces In Real Collisions Are
Not Constant
We treat the forces involved in a collision as though
they were constant, but in real-life situations the
forces will vary throughout the course of a
collision.
At any instant during a collision, the forces of the
two colliding objects will be equal and opposite.
When solving impulse problems, use the average
force over the time interval as the value for force.
Elastic and Inelastic Collisions
Objectives:
1. Identify different types of collisions.
2. Determine the changes in kinetic energy during
perfectly inelastic collisions.
3. Compare conservation of momentum and
conservation of kinetic energy in perfectly inelastic
and elastic collisions.
4. Find the final velocity of an object in perfectly
inelastic and elastic collisions.
Perfectly Inelastic Collisions
When two objects collide and move together as one
after the collision (stick together), the collision is a
perfectly inelastic collision.
The total momentum of the the two objects after the
collision is equal to the total momentum of the two
objects before the collision.
m1v1,i + m2v2,i = (m1 + m2)vf
Concept Check
A 1500 kg car traveling at 15.0 m/s to the south
collides with a 4500 kg truck that is initially at rest
at a stoplight. The car and truck stick together and
move together after the collision. What is the final
velocity of the two-vehicle mass?
Concept Check
3.8 m/s to the south
Concept Check
A grocery shopper tosses a 9.0 kg bag of rice into a
stationary 18.0 kg grocery cart. The bag hits the
cart with a horizontal speed of 5.5 m/s toward the
front of the cart. What is the final speed of the cart
and bag?
Concept Check
1.8 m/s
Concept Check
A 1.50 x 104 kg railroad car moving at 7.00 m/s to
the north collides with and sticks to another
railroad car of the same mass that is moving in the
same direction at 1.50 m/s. What is the velocity of
the joined cars after the collision?
Concept Check
4.25 m/s to the north
Concept Check
A dry cleaner throws a 22 kg bag of laundry onto a
stationary 9.0 kg cart. The cart and laundry bag
begins moving at 3.0 m/s to the right. Find the
velocity of the laundry bag before the collision.
Concept Check
4.2 m/s to the right
Concept Check
A 47.4 kg student runs down the sidewalk and
jumps with a horizontal speed of 4.20 m/s onto a
stationary skateboard. The student and skateboard
move down the sidewalk with a speed of 3.95 m/s.
Find the following:
a. the mass of the skateboard
b. how fast the student would have to jump to have
a final speed of 5.00 m/s.
Concept Check
3.00 kg
5.32 m/s
Kinetic Energy
Kinetic energy is not conserved in inelastic
collisions.
Some of the kinetic energy is converted to sound
energy and internal energy as objects are deformed
during the collision.
The decrease in kinetic energy can be calculated by
using the formula for kinetic energy.
Concept Check
A 0.25 kg arrow with a velocity of 12 m/s to the
west strikes and pierces the center of a 6.8 kg
target.
a. What is the final velocity of the combined mass?
b. What is the decrease in kinetic energy during the
collision?
Concept Check
a. 0.43 m/s to the west
b. 17 J
Concept Check
During practice, a student kicks a 0.40 kg soccer
ball with a velocity of 8.5 m/s to the south into a
0.15 kg bucket lying on its side. The bucket
travels with the ball after the collision.
a. What is the final velocity of the combined mass?
b. What is the decrease in kinetic energy during the
collision?
Concept Check
a. 6.2 m/s to the south
b. 3 J
Concept Check
A 56 kg ice skater travels at 4.0 m/s to the north
meets and joins hands with a 65 kg skater traveling
at 12.0 m/s in the opposite direction. Without
rotating, the two skaters continue skating together
with joined hands.
a. What is the final velocity of the two skaters?
b. What is the decrease in kinetic energy during the
collision?
Concept Check
a. 4.6 m/s to the south
b. 3.9 x 103 J
Elastic Collisions
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In an elastic collision two objects collide and return to their
original shapes with no loss of kinetic energy (they don't
stick together.)
– Both momentum and kinetic energy are
conserved.
In the real world, most collisions are not perfectly inelastic or
elastic but somewhere in between.
We consider all collisions in which the objects do not stick
together to be elastic.
Elastic Collisions
The total momentum is always constant throughout
the collision.
The total kinetic energy is also conserved.
m1v1,i +m2 v2,i = m1 v1,f + m2 v2.f
½ m1 v1,i2 + ½ m2 v2,i2 = ½ m1 v1,f 2+ ½ m2 v2,f 2
Concept Check
A 0.015 kg marble sliding to the right at 22.5 cm/s
on a frictionless surface makes an elastic head-on
collision with a 0.015 kg marble moving to the left
at 18.0 cm/s. After the collision, the first marble
moves to the left at 18.0 cm/s.
a. Find the velocity of the second marble after the
collision.
b. Verify your answer by calculating the total
kinetic energy before and after the collision.
Concept Check
a. 22.5 cm/s to the right
b. KEi = 6.2 x10-4 J = KEf
Concept Check
A 16.0 kg canoe moving to the left at 12.5 m/s
makes an elastic head-on collision with a 14.0 kg
raft moving to the right at 16.0 m/s. After the
collision, the raft moves to the left at 14.4 m/s.
Disregard any effects of the water.
a. Find the velocity of the canoe after the collision.
b. Verify your answer by calculating the total
kinetic energy before and after the collision.
Concept Check
a. 14.1 m/s to the right
b. 3.04 x 103 J