Chapter 6
Momentum and Collisions
Momentum

The linear momentum p of an
object of mass m moving with a
velocity v is defined as the product
of the mass and the velocity



p  mv
SI Units are kg m / s
Vector quantity, the direction of the
momentum is the same as the
velocity’s
Momentum components

p x  mv x and p y  mv y

Applies to two-dimensional motion
Impulse


In order to change the momentum of
an object, a force must be applied
The time rate of change of momentum
of an object is equal to the net force
acting on it


p
t

m(vf  vi )
t
 Fnet
Gives an alternative statement of Newton’s
second law
Impulse cont.

When a single, constant force acts
on the object, there is an impulse
delivered to the object

I  Ft

I is defined as the impulse

Vector quantity, the direction is the
same as the direction of the force
Impulse-Momentum
Theorem

The theorem states that the
impulse acting on the object is
equal to the change in momentum
of the object


Ft  p  m v f  m v i
If the force is not constant, use the
average force applied
Impulse Applied to Auto
Collisions

The most important factor is the
collision time or the time it takes
the person to come to a rest


This will reduce the chance of dying
in a car crash
Ways to increase the time


Seat belts
Air bags
Air Bags



The air bag increases
the time of the
collision
It will also absorb
some of the energy
from the body
It will spread out the
area of contact


decreases the
pressure
helps prevent
penetration wounds
Example 1
Calculate the magnitude of the linear momentum
for the following cases: (a) a proton with mass
1.67 × 10–27 kg, moving with a speed of 5.00 ×
106 m/s; (b) a 15.0-g bullet moving with a speed
of 300 m/s; (c) a 75.0-kg sprinter running with a
speed of 10.0 m/s; (d) the Earth (mass = 5.98 ×
1024 kg) moving with an orbital speed equal to
2.98 × 104 m/s.
Example 2
A 0.10-kg ball is thrown straight up into
the air with an initial speed of 15 m/s.
Find the momentum of the ball (a) at its
maximum height and (b) halfway to its
maximum height.
Example 3
A 75.0-kg stuntman jumps from a balcony
and falls 25.0 m before colliding with a pile
of mattresses. If the mattresses are
compressed 1.00 m before he is brought to
rest, what is the average force exerted by
the mattresses on the stuntman?
Example 4
A 0.500-kg football is thrown toward the east
with a speed of 15.0 m/s. A stationary
receiver catches the ball and brings it to rest
in 0.020 0 s. (a) What is the impulse
delivered to the ball as it’s caught? (b) What
is the average force exerted on the receiver?
Example 5
A force of magnitude Fx acting in the x-direction on a 2.00-kg
particle varies in time as shown in Figure P6.12. Find (a) the
impulse of the force, (b) the final velocity of the particle if it is
initially at rest, and (c) the final velocity of the particle if it is
initially moving along the x-axis with a velocity of
–2.00 m/s.
Example 6
A pitcher throws a 0.15-kg baseball so that
it crosses home plate horizontally with a
speed of 20 m/s. The ball is hit straight back
at the pitcher with a final speed of 22 m/s.
(a) What is the impulse delivered to the
ball? (b) Find the average force exerted by
the bat on the ball if the two are in contact
for 2.0 × 10–3 s.
Conservation of
Momentum

Momentum in an isolated system in
which a collision occurs is conserved



A collision may be the result of physical
contact between two objects
“Contact” may also arise from the
electrostatic interactions of the electrons in
the surface atoms of the bodies
An isolated system will have not external
forces
Conservation of
Momentum, cont

The principle of conservation of
momentum states when no
external forces act on a system
consisting of two objects that
collide with each other, the total
momentum of the system remains
constant in time

Specifically, the total momentum
before the collision will equal the total
momentum after the collision
Conservation of
Momentum, cont.

Mathematically:
m1v1i  m2 v2i  m1v1f  m2 v2f
 Momentum is conserved for the system of
objects
 The system includes all the objects
interacting with each other
 Assumes only internal forces are acting
during the collision
 Can be generalized to any number of
objects
Notes About A System


Remember conservation of
momentum applies to the system
You must define the isolated
system
Types of Collisions


Momentum is conserved in any collision
Inelastic collisions

Kinetic energy is not conserved


Some of the kinetic energy is converted into other
types of energy such as heat, sound, work to
permanently deform an object
Perfectly inelastic collisions occur when the
objects stick together

Not all of the KE is necessarily lost
More Types of Collisions

Elastic collision


both momentum and kinetic energy
are conserved
Actual collisions

Most collisions fall between elastic
and perfectly inelastic collisions
More About Perfectly
Inelastic Collisions


When two objects
stick together
after the collision,
they have
undergone a
perfectly inelastic
collision
Conservation of
momentum
becomes
m1v1i  m2 v 2i  (m1  m2 )v f
Some General Notes About
Collisions

Momentum is a vector quantity


Direction is important
Be sure to have the correct signs
More About Elastic
Collisions


Both momentum and kinetic
energy are conserved
Typically have two unknowns
m1v1i  m2 v 2i  m1v1f  m2 v 2 f
1
2

m1v 
2
1i
1
2
m2 v
2
2i

1
2
m1v 
Solve the equations
simultaneously
2
1f
1
2
2
m2 v 2 f
Summary of Types of
Collisions



In an elastic collision, both momentum
and kinetic energy are conserved
In an inelastic collision, momentum is
conserved but kinetic energy is not
In a perfectly inelastic collision,
momentum is conserved, kinetic energy
is not, and the two objects stick
together after the collision, so their final
velocities are the same
Problem Solving for One Dimensional Collisions

Coordinates: Set up a coordinate
axis and define the velocities with
respect to this axis


It is convenient to make your axis
coincide with one of the initial
velocities
Diagram: In your sketch, draw all
the velocity vectors and label the
velocities and the masses
Problem Solving for One Dimensional Collisions, 2

Conservation of Momentum:
Write a general expression for the
total momentum of the system
before and after the collision


Equate the two total momentum
expressions
Fill in the known values
Problem Solving for One Dimensional Collisions, 3

Conservation of Energy: If the
collision is elastic, write a second
equation for conservation of KE, or
the alternative equation


This only applies to perfectly elastic
collisions
Solve: the resulting equations
simultaneously
Sketches for Collision
Problems


Draw “before”
and “after”
sketches
Label each object


include the
direction of
velocity
keep track of
subscripts
Sketches for Perfectly
Inelastic Collisions



The objects stick
together
Include all the
velocity directions
The “after”
collision combines
the masses
Glancing Collisions

For a general collision of two objects in
three-dimensional space, the
conservation of momentum principle
implies that the total momentum of the
system in each direction is conserved

m1v1ix  m2 v 2ix  m1v1f x  m2 v 2 f x and
m1v1iy  m2 v 2iy  m1v1f y  m2 v 2 f y

Use subscripts for identifying the object,
initial and final velocities, and components
Glancing Collisions



The “after” velocities have x and y
components
Momentum is conserved in the x direction and
in the y direction
Apply conservation of momentum separately
to each direction
Problem Solving for TwoDimensional Collisions

Coordinates: Set up coordinate
axes and define your velocities
with respect to these axes


It is convenient to choose the x- or yaxis to coincide with one of the initial
velocities
Draw: In your sketch, draw and
label all the velocities and masses
Problem Solving for TwoDimensional Collisions, 2


Conservation of Momentum: Write
expressions for the x and y components
of the momentum of each object before
and after the collision
Write expressions for the total
momentum before and after the
collision in the x-direction and in the ydirection
Problem Solving for TwoDimensional Collisions, 3

Conservation of Energy: If the
collision is elastic, write an
expression for the total energy
before and after the collision



Equate the two expressions
Fill in the known values
Solve the quadratic equations

Can’t be simplified
Problem Solving for TwoDimensional Collisions, 4

Solve for the unknown quantities



Solve the equations simultaneously
There will be two equations for
inelastic collisions
There will be three equations for
elastic collisions
Example 7
A rifle with a weight of 30 N fires a 5.0g bullet with a speed of 300 m/s. (a)
Find the recoil speed of the rifle. (b) If a
700-N man holds the rifle firmly against
his shoulder, find the recoil speed of the
man and rifle.
Example 8
A 75.0-kg ice skater moving at 10.0 m/s crashes
into a stationary skater of equal mass. After the
collision, the two skaters move as a unit at 5.00
m/s. Suppose the average force a skater can
experience without breaking a bone is 4 500 N. If
the impact time is 0.100 s, does a bone break?
Example 9
A 0.030-kg bullet is fired vertically at 200
m/s into a 0.15-kg baseball that is initially
at rest. How high does the combined bullet
and baseball rise after the collision,
assuming the bullet embeds itself in the
ball?
Example 10
A 1 200-kg car traveling initially with a speed of 25.0 m/s in
an easterly direction crashes into the rear end of a 9 000-kg
truck moving in the same direction at 20.0 m/s (Fig. P6.32).
The velocity of the car right after the collision is 18.0 m/s to
the east. (a) What is the velocity of the truck right after the
collision? (b) How much mechanical energy is lost in the
collision? Account for this loss in energy.
Example 11
A 12.0-g bullet is fired horizontally into a 100-g
wooden block that is initially at rest on a
frictionless horizontal surface and connected to a
spring having spring constant 150 N/m. The bullet
becomes embedded in the block. If the bullet–
block system compresses the spring by a maximum
of 80.0 cm, what was the speed of the bullet at
impact with the block?
Example 12
A billiard ball rolling across a table at 1.50 m/s makes a
head-on elastic collision with an identical ball. Find the
speed of each ball after the collision (a) when the second
ball is initially at rest, (b) when the second ball is moving
toward the first at a speed of 1.00 m/s, and (c) when the
second ball is moving away from the first at a speed of 1.00
m/s.
Example 13
A 90-kg fullback moving east with a speed of 5.0 m/s is
tackled by a 95-kg opponent running north at 3.0 m/s. If
the collision is perfectly inelastic, calculate (a) the velocity
of the players just after the tackle and (b) the kinetic
energy lost as a result of the collision. Can you account for
the missing energy?