# Relativistic Kinetic Energy Calculator

Created by Bogna Szyk
Reviewed by Małgorzata Koperska, MD and Steven Wooding
Last updated: Aug 22, 2022

If you're analyzing an object traveling at high speed, you will probably observe that the results provided by our regular kinetic energy calculator are not entirely accurate. In such a case, make sure to use this relativistic kinetic energy calculator that takes into consideration relativistic effects.

## What is the kinetic energy?

Kinetic energy, by definition, is the energy resulting from the motion of an object. For example, a car traveling along a highway has certain energy – if it hits another vehicle, the outcome will be much more destructive than if it were moving at 5 mph.

For objects moving at low speeds, you can determine the kinetic energy using the following formula:

$\small \text{KE} = \frac{1}{2}mv^2$

where:

• $\text{KE}$ – Kinetic energy expressed in joules;
• $m$ – Mass of the object; and
• $v$Speed of the object.

## Relativistic effects in measuring the kinetic energy

If the object moves at high velocity – at least 1% of the speed of light – relativistic effects begin to be noticeable. Due to that fact, you can no longer use the simplified formula mentioned in the previous section.

In such a case, you must calculate the kinetic energy from the famous Einstein's E = mc² equation. This equation describes the total (i.e., both potential and kinetic) energy of the body. To obtain the relativistic kinetic energy, you need to subtract the potential energy (previously found with the potential energy formula) from the total energy:

$\small\rm KE = E - PE$

The potential energy, in this case, is equal to the rest-mass energy:

$\small \begin{split} \text{PE} &= m_0c^2\\[.5em] \text{KE} &= mc^2 - m_0c^2 \end{split}$

where:

• $c$ – Speed of light – a constant equal to 299,792,458 m/s; and
• $m_0$ is the rest mass of the object.

Rest mass $m_0$ differs from the regular mass $m$. $m_0$ describes the mass of the object at rest – that is, if it's not moving. If the object moves, its true mass increases according to this formula:

$\small m = \frac{m_0}{\sqrt{1 - v^2/c^2}}$

Combining all of these equations, you can derive the relativistic kinetic energy equation:

$\small \text{KE} = m_0c^2 \left(\sqrt{1 - v^2/c^2} - 1\right)$

## Other relativity effects

If you're interested in relativity, check out the velocity addition and electron speed calculators!

Bogna Szyk
Mass
lb
Velocity
mph
Kinetic energy
MJ
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