# Relativistic Kinetic Energy Calculator

Created by Bogna Szyk
Reviewed by Małgorzata Koperska, MD and Steven Wooding
Last updated: Jun 05, 2023

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 Einstein's famous 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 calculator and the electron speed calculator.

## FAQ

### What is the relativistic formula for kinetic energy?

The relativistic kinetic energy is given by KE = m0c2(√(1 - v2/c2) - 1), where m0 is rest mass, v is velocity, and c is the speed of light. This formula takes into account both the total rest mass energy and kinetic energy of motion.

### Can relativistic kinetic energy be negative?

No, the relativistic kinetic energy cannot be negative. It is always greater than or equal to zero. The square of the velocity of an object is always positive, so the mass would have to be negative, which has never been observed. Therefore, kinetic energy can never be negative.

### How can I calculate the kinetic energy of an electron with a velocity of 0.990c?

To calculate the kinetic energy of an electron, follow these steps:

1. Find the mass of an electron: 9.11 × 10−31 kg.

2. Put the data in the formula:

KE = m0c2(√(1 - v2/c2) - 1)

KE = 9.11 × 10−31 kg × (3.00 × 108)2 (√(1 - 0.990c2/c2) - 1)

3. Note that c will reduce in the formula under the square root.

4. Calculate kinetic energy as 4.98 × 10−13 J.

### What will be the classical value for kinetic energy at velocity 0.990c?

4.02 × 10−14 J. Enter the mass of an electron 9.11 × 10−31 kg and the speed of light 3.00 × 108 into the classical (non-relativistic) formula:

KE = 1/2mv2
KE = 1/2 × 9.11 × 10−31 kg × (0.990)2 × (3.00 × 108 m/s)2
KE = 4.02 × 10−14 J

Note that with a velocity of 99.0% of the speed of light, only the relativistic value gives the correct results.

### What happens to relativistic kinetic energy at low velocities?

At low velocities, the relativistic kinetic energy formula will reduce to classical kinetic energy. If the velocity v is significantly less than the speed of the light c, the expression (√(1 - v2/c2) - 1) is simplified to the form 1v2/2c2. Entering this into the expression for relativistic kinetic energy gives you:

KE = mc2(1v2/2c2) = 1/2mv2.

### Can an object can reach the speed of light?

Unfortunately, no. No object with mass can attain the speed of light, and this has been confirmed in numerous experiments. No matter how much energy you supply to a particle, its velocity can only approach the speed of light but never reach it.

Bogna Szyk
Mass
lb
Velocity
mph
Kinetic energy
MJ
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