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**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:

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

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 m_{0}
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.