Conservation of Momentum

A stone hits a ball and bounces off. The total momentum before and after collision are the same. This is called the conservation of momentum.

Ok, that is not the whole story. So lets start again.

We shall look at the simple case of movement along a straight line. A stone and a ball sliding along a perfectly smooth floor hit each other. Lets look at the ball for now. The stone applies a force F on the ball of mass m during the contact. The contact time t is usually short, but enough to change the momentum of the ball.

We already know how F and t are related to the momentum change. Newton's second law says that

F = (mv - mu)/t,

where u is the initial velocity of the ball and v the final velocity. Multiply both sides by t and we get

Ft = mv - mu.

The right hand side is the change in momentum. The left hand side Ft is called the impulse. We have a simple formula:

change of momentum = force x time.

Lets return to the stone hitting the ball. During the contact time t, the stone applies a force F on the ball and changes it momentum by Ft.

According to Newton's third law, the ball applies an equal and opposite force on the stone -F. So the momentum of the stone would be changed by -Ft. This means that if the momentum of the ball is increased, the momentum of the stone must be decreased by the same amount.

Therefore, the total momentum must remain the same after the collision!

We have thus derived the principle of conservation of momentum in a very simple way. Since we often need to do calculations on this, it would be useful to write down a formula for this. Lets make some symbols:

mA = mass of stone
uA = initial velocity of stone
vA = final velocity of stone

mB = mass of ball
uB = initial velocity of ball
vB = final velocity of ball

The total momentum before collision = mAuA + mBuB

The total momentum after collision = mAvA + mBvB

Since they are equal, we have

mAuA + mBuB = mAvA + mBvB

This is the formula.




Copyright 2011 by Kai Hock. All rights reserved.
Last updated: 19 May 2011