# 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*!

## 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**:

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:

🙋 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:

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:

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: