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Velocity Addition Calculator

Created by Miłosz Panfil, PhD
Reviewed by Bogna Szyk and Steven Wooding
Last updated: Jan 18, 2024


This velocity addition calculator will help you better understand special relativity and the relativistic velocity addition formula. You might have heard that the velocities cannot be just added. Instead, we should use the Einstein velocity addition equation, sometimes called also relativistic velocity addition equation.

Interested in relativity? Check out the relativistic kinetic energy calculator or electron speed calculator!

Speed of light is the same for everyone

Imagine your friend flying a spaceship at high speed and fires a projectile. What is the speed of the projectile? If the spaceship's speed is vv and your friend fired the projectile at speed ww (with respect to the spaceship), then it seems logical that you will see the projectile traveling at speed v+wv + w.

However, this cannot be true. If your friend turns on the laser instead of firing a projectile, then the speed of light he observes is, well, the speed of light cc. The speed of light is the same for every observer, which means that you will see the speed of light is also cc, not v+cv + c. This argument shows that we cannot just add velocities.

This is where our relativistic velocity addition calculator can help resolve this contraction. Keep reading to find out more.

Time and space in the special relativity

Both you and your friend observe the same speed of light because when moving, time passes at different rates, and distances seem different. These two effects are time dilation and length contraction. You can see how their work using our time dilation calculator and length contraction calculator. If we combine these two effects, the result is the relativistic velocity addition formula.

Relativistic velocity addition formula

How do we add the velocities? The formula is:

u=v+w1+(vw/c2),\small u = \frac{v + w}{1 + (vw/c^2)},

where:

  • uu – Speed of the projectile as seen outside of the spaceship;
  • vv – Speed of the spaceship;
  • ww – Speed of the projectile as seen from the spaceship; and
  • cc – Speed of light (299,792,458 m/s).

When the speed of the spaceship or the projectile's speed is slow compared to the speed of light, the formula reduces to the sum of the velocities u=v+wu = v + w.

However, if the velocities vv and ww are both large, then the projectile's speed, as seen outside of the spaceship, is much lower than their sum v+wv + w and never exceeds the speed of light. In the extreme case, when one of the velocities is equal to the speed of light cc, the other one won't exceed it, either; the speed of light is the same for everyone.

You can check other effects of special relativity with the famous E = mc² calculator.

FAQ

How do I use the velocity addition formula?

To use the velocity addition formula:

  1. Take A as the observer and B and C as moving objects.

  2. Find the velocity of B as seen by A, v, and the velocity of C with respect to B, w.

  3. The Galilean transformation is v + w.

  4. For a relativistic velocity addition, divide the Galilean result by 1 + ((v × w)/c²).

What velocities are relativistic?

Relativistic velocities are high enough for us to observe the effects of special relativity. In practice, this means considering objects at speeds, v, close to that of light, c, as only then the Lorentz factor, γ = √(1 - v²/c²), isn't negligible.

What is the speed of an electron emitted by a nucleus?

The speed of the electron is ~0.72 c if we assume that the nucleus is moving at 0.35 c with respect to the lab, and the electron's velocity is 0.5 c, relative to its parent. c is the speed of light. This is the result of the relativistic velocity addition.

Can you use the velocity formula for all components?

In short, no. The most common form of relativistic velocity addition is for motion along the x-axis. If you derive it for the y- and z-components, you'll see that the formulae are different, e.g., they contain the Lorentz factor, γ = √(1 - v²/c²).

Miłosz Panfil, PhD
Spaceship velocity (v)
c
Projectile velocity (w)
c
Relative projectile velocity (u)
c
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