Why invent the momentum concept in the first place?

A brick hanging in space has a firework rocket attached to it. The rocket burns and makes the brick accelerate up to a certain speed, then the firework expires. It is important to have a measure of the 'smashing power' of the brick – how many pains of glass could it plough through in virtue of its speed?

Assume the firework applies a constant force. There are two simple measures of smashing power:

  1. Force of the rocket multiplied by the distance the brick is pushed through during the burn. (Force x distance).

  2. Force of the rocket multiplied by the time the rocket burns. (Force x time).

The first is called 'energy', more specifically kinetic energy and the second is momentum. (If the force of the rocket varied you would have to integrate rather than simply multiply, but it amounts to the same thing.)

Both quantities are conserved. That is their total quantity does not change.

An instant qualification is necessary.

Imagine two balls of wet clay collide. Kinetic energy is not unchanged before and after, but the balls have warmed slightly, and if this 'heat' energy is added in, the total energy before and after is indeed unaltered.

It is not the same for momentum. It is just unchanged full stop. There is no heat analogy for momentum. (It is tempting to argue that there is, it is just so slight it does not notice.)

p = mv

If force x time or the equivalent integral is evaluated it works out as p = mv. This is true in relativity just as it is in Newtonian mechanics. Some of the above nodes could perhaps slightly mislead in this respect.

Nothing may be accelerated beyond the velocity of light in a vacuum, c, so to prevent this an object's mass – the proportionality factor that measures its resistance to acceleration (force is proportional to acceleration) – must increase as it goes faster.

So if the thing has mass m0 when it is at rest relative to the measuring apparatus its mass increases as it goes faster according to the formula given in the above pieces. Its momentum is always its mass, m, multiplied by velocity.