# Relativistic Kinetic Energy Calculator

Created by Luciano Mino
Last updated: Oct 09, 2022

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!

## Special relativity and its implications

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.

🔎 For the following section, we will assume you're already familiar with these topics and will only give a brief explanation of the concepts.

## Relativistic kinetic energy formula

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

$K = E - mc^{2}$

equivalently:

$K = \frac{mc^{2}}{\sqrt{1-\frac{v^{2}}{c^{2}}}} - mc^{2}$

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:

$K \thickapprox \frac{1}{2}\frac{v^{2}}{c^{2}}mc^{2} = \frac{1}{2}mv^{2}$

🙋 The term 1 / √(1-v²/c²) is called the Lorentz factor. Read more about it in our lorentz factor calculator.

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.

## Using this tool as a relativistic momentum calculator

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

$m^{2}c^{2} = \frac{E^{2}}{c^{2}} - p^{2}$

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 is Einstein's famous E = mc². Read more about it in our E = mc² calculator!

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:

$E=\sqrt{p^{2}c^{2}+m^{2}c^{4}}$

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

$\frac{K^{2}}{c^{2}}+2Km = p^{2}$

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:

$p = \frac{mv}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$
Luciano Mino
Mass
lb
Velocity
mph
Kinetic energy
MJ
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