Suppose you know the mass **m** of a particle, and you are
given its momentum **p**.
If the particle is moving at high speed, then its momentum
is related to its velocity **v** like so:

If you try to solve for the velocity **v**,
you might think yourself stumped after the first step:

"Oh noes!" you might say,
"I can't isolate the velocity **v**,
because **gamma** has the velocity inside it."

Don't despair. You CAN still isolate the velocity; you just need to do a bit of algebra. Let me lead the way.

It would be annoying to have to keep writing
**p/m** in all the following steps,
so I'll define a single variable to hold that
ratio of momentum to mass.

So our problem now looks like this:

A good first step is to square both sides,
so that we get rid of the square root inside the expression
for **gamma**.

Now, let's multiply both sides by **(1 - v^2/c^2).**

We can put all the terms with **v** together on the left ...

... and then isolate the velocity **v.**

We end up with an expression for velocity of a relativistic particle,
given its mass **m** and momentum **p**:

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.