Relativistic Kinetic Energy Calculator

With this relativistic kinetic energy calculator, you will be able to test and understand special relativity's results when applied to a fast-moving object's kinetic energy.

In this short text below, we will cover:

• The postulates of special relativity;
• The relativistic kinetic energy formula or kinetic energy formula near the speed of light;
• How to use this tool as a relativistic momentum calculator using the energy-momentum relation; and
• More about Einstein's theory of special relativity!

Einstein developed the special theory of relativity in 1905 in his paper "On the electrodynamics of moving bodies". In it, he proposed two postulates:

• That the laws of physics are the same in all inertial frames of reference; and
• Light propagates in empty space with a constant speed $c$, independent of the state of motion of the emitting body.

These two simple postulates have many different and counterintuitive results, such as length contraction, time dilation, and velocity addition.

In special relativity, we define the kinetic energy of a body in terms of its total energy and its rest energy:

equivalently:

where:

• $m$ is the rest mass of the body, i.e., the mass when the body is at rest;
• $v$ is the body's velocity; and
• $c$ is the speed of light, $c = 299,792,458\ \text{m/s}$

At low velocities, if we use a binomial expansion for $\sqrt{1-\frac{v^{2}}{c^{2}}}$ and replace it into the expression above, we regain the classical expression:

Give it a try! Use the relativistic kinetic energy calculator to compare the relativistic kinetic energy of a body near the speed of light against its kinetic energy according to the classical formula.

The momentum-energy four-vector possesses an invariant quantity under a Lorentz transformation, the invariant mass of the system:

where:

• $E$ is the total energy of the system.
• $p$ is the system's total momentum.
• $m$ is the invariant mass. While energy and momentum may differ, all observers agree on the invariant mass.

This relation allows our relativistic kinetic energy calculator to help you obtain other parameters from special relativity.

If we wanted to use this tool as a relativistic momentum calculator, we should rearrange the equation above to find an expression for energy first:

Then, we can replace it in the relativistic kinetic energy equation and solve for $p$ to get:

Alternatively, you can type in the numbers for the kinetic energy and mass of the body, and our relativistic kinetic energy calculator will automatically tell you its velocity. Then, you can use the velocity to find the relativistic momentum with the following expression: