Chapter 8
Momentum and Momentum
Conservation
Momentum

The linear momentum p of an object
of mass m moving with a velocity is
defined as the product of the mass
and the velocity v
•
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, e.g.
v  vo  at
•
mv  mvo  mat  mvo  Ft
Ft  mv  mvo
• Gives an alternative statement of Newton’s
second law
Impulse cont.

When a single, constant force acts on
the object, there is an impulse J
delivered to the object
•
J  Ft  F (t 2  t1)
• is defined as the impulse
• Vector quantity, the direction is the
same as the direction of the force
• Unit N·s=kg·m/s
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 2  t1)  p 2  p1
• Impulse=change in momentum
(vector!)
• 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
Conservation of Momentum

Total momentum of a system equals
to the vector sum of the momenta
p   pi  p1  p 2  p3  ...
i

When no resultant external force
acts on a system, the total
momentum of the system remains
constant in magnitude and direction.
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:
mAvA  mB vB  mAv'A  mBv'B
• 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
Example
A 60 grams tennis ball traveling at
40m/s is returned with the same
speed. If the contact time between
racket and the ball is 0.03s, what is
the force on the ball?
Example
A 5kg ball moving at 2 m/s collides
head on with another 3kg ball
moving at 2m/s in the opposite
direction. If the 3kg ball rebounds
with the same speed. What is the
velocity of the 5 kg ball after
collision?
Recoil


System is released from rest
Momentum of the system is zero
before and after
mAv  mBv  0
'
A
'
B
Example
4 kg rifle shoots a 50 grams bullet. If
the velocity of the bullet is 280 m/s,
what is the recoil velocity of the rifle?
Types of Collisions


Momentum is conserved in any
collision
Perfect elastic collision
• both momentum and kinetic energy are
conserved
KEbefore  KEafter

Collision of billiard balls, steel balls
More Types of Collisions

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
• completely inelastic collisions occur when the
objects stick together


Not all of the KE is necessarily lost
Actual collisions
• Most collisions fall between elastic and
completely 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
m Av A  mB vB  (m A  mB )V
m A v A  mB v B
V
m A  mB
Example
Railroad car (10,000kg) travels at
10m/s and strikes another railroad
car (15,000kg) at rest. They couple
after collision. Find the final velocity
of the two cars. What is the energy
loss in the collision?
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
mAv A  mB vB  mAv' A  mB v'B

1
1
1
1
2
2
'2
'2
mAv A  mB vB  mAv A  mB vB
2
2
2
2
Solve the equations simultaneously
A Simple Case, vB=0
Head on elastic collision with object B
at rest before collision.
One can show
m A  mB
vA 
vA
m A  mB
2m A
vB 
vA
m A  mB
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
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
Ballistic Pendulum



Measure speed of bullet
Momentum conservation of the
collision
Energy conservation during the
swing of the pendulum