# Video: Relativistic Kinetic Energy Relation to Velocity

What is the velocity of an electron that has a kinetic energy of 0.750 MeV?

02:27

### Video Transcript

What is the velocity of an electron that has a kinetic energy of 0.750 MeV?

We can call the given kinetic energy of this electron capital KE. And we want to solve for this electronβs velocity, which weβll call π£. To start out on our solution, letβs recall the mathematical relationship for relativistic kinetic energy.

An objectβs relativistic kinetic energy equals the quantity πΎ minus one times rest mass times the speed of light squared, where πΎ is defined as one over the square root of one minus π£ squared over π squared. So the kinetic energy of the electron is equal to πΎ minus one times the rest energy of the electron: its rest mass times π squared.

Notice that π£, the velocity of the electron, is in this equation. Itβs that value we want to solve for. If we divide both sides of the equation by π sub zero π squared and add one to both sides, we see that KE over π sub zero π squared plus one equals πΎ.

Rearranging further, we find that one minus π£ squared over π squared equals one over KE divided by π sub zero π squared plus one quantity squared. And finally, we can solve for π£, the velocity, which equals the speed of light π times the square root of this expression in brackets.

Looking at this expression, we see weβve been given the kinetic energy of the electron in the problem statement. The rest energy of the electron on the other hand π sub zero π squared is something we want to look up in a table. When we do, we find that the rest energy of an electron is equal to 0.511 mega electron volts.

So weβre now prepared to plug in and solve for the electron velocity π£. When we do inserting values for the electrons kinetic energy and its rest energy, we find that this expression multiplying the speed of light π is equal to 0.914. This means that the speed of the electron with this kinetic energy is 0.914 times the speed of light.