Lessons 6-8 notes - Conservation of Momentum
Objectives
Be able to state the principle of conservation of momentum;
Be able to apply the principle of conservation of momentum to solve problems
when bodies interact in one dimension;
Be able to define a perfectly elastic collision and an inelastic collision;
Be able to explain that whilst the momentum of a system is always conserved
in the interaction between bodies, some change in kinetic energy usually
occurs.
Outcomes
Be able to state the principle of conservation of momentum.
To understand the difference between elastic and inelastic collisions.
Be able to apply the principle of conservation of momentum to solve a number
of problems when different types of bodies interact in one dimension.
To be able to explain the difference between elastic and inelastic collisions.
To be able to give examples of elastic and inelastic collisions.
To be able to apply the principle of conservation of momentum to explosions
in one dimension.
Conservation of momentum
The momentum of objects is constant, unless there’s a change in either the
mass or velocity, or both the mass and velocity change. This is known as the
Law of Conservation of Momentum, which says:
The total momentum of a group of objects remains constant unless
outside forces act on the objects. Momentum can be transferred from
one or more objects to other objects, but the total momentum remains
the same.
ie. The momentum before a collision or explosion = the momentum after
the collision or explosion.
This is very important in situations that involve collisions between objects, as
in auto accidents, games like bowling and pool, and contact sports like
football, rugby and basketball.
Momentum can also be transferred from one object to another as in golf,
when some of the momentum of a swinging golf club gets transferred to the
stationary golf ball, resulting in the golf ball gaining momentum. This causes a
change in the velocity of the golf ball.
Unless there is a change in mass of one or more objects during a collision,
the only factor that would change is the velocity. So during some collisions, if
there is no change in the speed of an object, there must be a change in the
direction of the object or objects.
The total linear momentum remains constant even if the kinetic energy of the
system changes.
If two skaters initially at rest push each other, they both will begin to move.
The kinetic energy of the system has changed due to work done by internal
forces. The total momentum of the system, however must remain equal to
zero. We can ignore gravity in this case since there is no work done by
gravity. We are also ignoring friction.
Collisions
During a collision, kinetic energy may be converted
to another form such as heat or used to
permanently deform one or both of the objects
involved. So the kinetic energy before the collision
is greater than the kinetic energy afterwards. This
type of collision is called an inelastic collision and
conservation of energy within the system does not
apply.
Most collisions in real life are inelastic since there
will always be some loss of kinetic energy due to
friction. Collisions in which no kinetic energy is
converted into another form are said to be elastic.
The conservation of energy equation can then be
applied to the system.
For both types of collision, momentum is
conserved.
Conservation of Momentum Analysis
We have said that:
The momentum before a collision or explosion = the momentum after
the collision or explosion.
Take a look at the diagram below:
+
u1
u2
m1
v1
m2
Before collision
v2
m1
m2
After collision
If a mass m1 moving at a velocity u1 collides with a mass m2 moving at a
velocity u2 such that after the collision m1 moves at v1 and m2 moves at v2 by
the conservation of momentum we can say that:
The momentum before a collision = the momentum after the collision
m1 u1 + m2 u2 = m1 v1 + m1 v2
Example
A 120kg rugby forward is moving west at 2 m/s and tackles an 80 kg winger
moving east at 8 m/s. After the collision, both players move east at what
speed?
Here is a methodical approach to answering collision questions
+
1 Draw a Diagram:
u1
u2
v
m1
Before collision
m2
m1
m2
After collision
2 Collect the data
m1 = 80kg; m2 = 120kg; u1 = 8 m/s; u2 = -2m/s (since the object is travelling in
a negative direction)
3 State the principle of conservation of momentum
By the conservation of momentum;
The momentum before a collision = the momentum after the collision
m1 u1 + m2 u2 = m1 v1 + m1 v2
This becomes:
m1 u1 + m2 u2 = (m1+ m2)v (since v1 and v2 are the same since the objects
move off together)
4 Put in the numbers
(80x8) + (120x(-2)) = (80+120)xv
640 – 240 = 200v
400 = 200v
5 Rearrange for v
v=400/200
v=2
Therefore: v = 2m/s (travelling in the positive direction (or east in this
example))
Extension
An explosion is a system where there is zero momentum before the event and
zero after. (since momentum is conserved). However the kinetic energy is
greater directly after the explosion.
Examples may be as simple as two ice skaters pushing away from each
other; or two spring loaded trucks doing the same. Or the examples could be
guns and stuff…
Example
A gun of mass 200 kg fires a 4 kg shell at 150 ms-1. What is the recoil velocity
of the gun?
Use the method above to work out the velocity
Here is a brief answer, check you can get it.
Momentum before explosion = Momentum after explosion
0 = 200 x v + 4x150
so v = - 600/200
Therefore: v = 3 ms-1