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Relativistic Velocity Addition Calculator

Calculates the relative velocity of objects moving at a significant fraction of the speed of light.

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Resultant Velocity

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Overview

The Relativistic Velocity Addition formula determines the velocity of an object as perceived by an observer in a different inertial frame moving at a constant speed. It is a core component of Special Relativity, ensuring that the combined speed of two objects never exceeds the universal speed limit of light.

Symbols

Variables

u = Resultant Velocity, v = Frame Velocity, u' = Relative Velocity, c = Speed of Light

Resultant Velocity
m/s
Frame Velocity
m/s
u'
Relative Velocity
m/s
Speed of Light
m/s

Apply it well

When To Use

When to use: Apply this formula when objects are moving at relativistic speeds, typically exceeding 10% of the speed of light, relative to an observer. It is used when a simple addition of velocities in classical mechanics would inaccurately suggest a speed close to or beyond the speed of light.

Why it matters: This equation preserves the second postulate of special relativity: that the speed of light is constant for all observers regardless of their motion. It is practically applied in high-energy physics to track subatomic particles and in the timing corrections required for global satellite navigation systems.

Avoid these traps

Common Mistakes

  • Applying simple Newtonian addition (v + u') for particles moving at high fractions of c.
  • Incorrectly squaring the entire denominator instead of just the speed of light term.
  • Mismatched units between the velocities and the speed of light constant.

One free problem

Practice Problem

A spacecraft moves at 0.6c relative to a planet. It launches a probe forward at 0.5c relative to the spacecraft's own frame. What is the velocity of the probe as measured by an observer on the planet?

Frame Velocity0.6 m/s
Relative Velocity0.5 m/s
Speed of Light1 m/s

Solve for:

Hint: Express velocities as decimals where c = 1 to simplify the denominator calculation.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Halliday, Resnick, Walker, Fundamentals of Physics
  2. Wikipedia: Relativistic velocity addition
  3. NIST CODATA
  4. Wikipedia: Speed of light
  5. Halliday, Resnick, Walker Fundamentals of Physics
  6. Britannica Special relativity
  7. Wikipedia article Special relativity
  8. University Physics — Special Relativity (Griffiths / Helliwell)