Created by Luis Hoyos
Last updated: Sep 24, 2022

When calculating velocity addition in relativistic scenarios, something changes compared to classical mechanics: we cannot just add the velocities; we must consider the effects of special relativity for speeds comparable to the speed of light.

With this special relativity calculator, you can obtain any of the relativistic velocity addition formula variables. Don't you know why you cannot just add velocities? Keep reading this article to discover it.

## Why do we need to calculate relativistic velocity addition?

When studying classical mechanics and dealing with different inertial reference frames, we calculate the relative velocities between those frames using Galilean relativity.

For example, imagine that a spaceship travels with a speed of $v$ (relative to you), and it fires a projectile with a speed of $w$ (relative to itself). According to Galilean relativity and common sense, the projectile has to travel at speed $v + w$ relative to you.

But what if the spaceship turned on a laser (whose speed equals the speed of light, $c$) and not a projectile? According to the Galilean analysis, the light would travel with a speed $v + c$ relative to you, violating the second postulate of special relativity: the speed of light in a vacuum is the same in all inertial frames, independently of the motion of the source. The reason behind this postulate is that time passes at different rates for reference frames moving relative to each other. As with velocity, this postulate also has effects when calculating time intervals (see our time dilation calculator) and lengths (see our length contraction calculator).

The previous example lets us note some points:

• We cannot just add velocities. We must consider the effects of special relativity if calculating velocity-addition (formula above).
• Common sense doesn't always work.

Now, let's look at the formula for relativistic velocity addition calculation.

## Relativistic velocity formula

To calculate the relativistic velocity addition, we use the following formula:

$u = \frac{v + w}{1 + vw/c^2}$

, where:

• $u$ – Speed of the projectile as seen outside of the spaceship;
• $v$ – Speed of the spaceship;
• $w$ – Speed of the projectile as seen from the spaceship; and
• $c$ – Speed of light (299,792,458 m/s).

Like relativistic kinetic energy, the relativistic velocity gives the same result as its Galilean counterpart when $v$ or $w$ is small compared to the speed of light. If that's not the case, the speed of the projectile seen outside of the spaceship ($u$) is lower than $v + w$.

Luis Hoyos
Spaceship velocity (v)
c
Projectile velocity (w)
c
Relative projectile velocity (u)
c
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