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 electron’s velocity, which we’ll call 𝑣. To start out on our solution, let’s recall the mathematical relationship for relativistic kinetic energy.
An object’s relativistic kinetic energy equals the quantity 𝛾 minus one times rest mass times the speed of light squared, where 𝛾 is defined as one over the square root of one minus 𝑣 squared over 𝑐 squared. So the kinetic energy of the electron is equal to 𝛾 minus one times the rest energy of the electron: its rest mass times 𝑐 squared.
Notice that 𝑣, the velocity of the electron, is in this equation. It’s that value we want to solve for. If we divide both sides of the equation by 𝑚 sub zero 𝑐 squared and add one to both sides, we see that KE over 𝑚 sub zero 𝑐 squared plus one equals 𝛾.
Rearranging further, we find that one minus 𝑣 squared over 𝑐 squared equals one over KE divided by 𝑚 sub zero 𝑐 squared plus one quantity squared. And finally, we can solve for 𝑣, the velocity, which equals the speed of light 𝑐 times the square root of this expression in brackets.
Looking at this expression, we see we’ve been given the kinetic energy of the electron in the problem statement. The rest energy of the electron on the other hand 𝑚 sub zero 𝑐 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 we’re now prepared to plug in and solve for the electron velocity 𝑣. When we do inserting values for the electrons kinetic energy and its rest energy, we find that this expression multiplying the speed of light 𝑐 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.