A Roller Coaster Ride through Relativity/Appendix G

The relation between Energy and Momentum
The total relativistic energy E and the relativistic momentum p of a body are given by the following expressions:

We wish to eliminate v from these equations. First square and multiply across:

Now for a diabolically cunning move, multiply the second equation by c2 and subtract!

from which we obtain:

An alternative (and in my opinion better) way of writing this equation is:

where E0 is the rest-mass energy of the body.

It is instructive to compare this expression with the non-relativistic relation between energy and momentum which is calculated as follows

so

It is not easy to see, at first, how the relativistic expression will reduce (as it must) to the non-relativistic one when v is small, but it does. Watch!

Since

we can write

Now (E - E0) is just the relativistic kinetic energy KEr which, at low speeds approximates to the ordinary kinetic energy KE.

At low speeds, the total relativistic energy E and the rest-mass energy E0 are virtually equal and equal to Mc2 so:

from which it is easy to see that

as expected.

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