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#1 Free Velocity Addition Calculator (Relativistic) 2025

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Relativistic Velocity Addition Calculator

Adding Velocities in Special Relativity

In classical (Newtonian) physics, if you are on a train moving at velocity u and you throw a ball forward at velocity v' relative to the train, an observer on the ground would see the ball moving at u + v'. However, this simple addition breaks down at speeds approaching the speed of light (c).

Einstein's Special Theory of Relativity provides a different formula for adding velocities. This formula ensures that the resultant velocity never exceeds the speed of light, which is a fundamental postulate of the theory.

How to use this calculator:

  1. Enter Velocity u: This is the velocity of a moving reference frame S' (e.g., a spaceship) relative to a stationary frame S (e.g., Earth).
  2. Enter Velocity v': This is the velocity of an object (e.g., a probe, or light pulse) relative to the moving frame S' (the spaceship).
  3. Velocities can be positive (in the chosen direction) or negative (opposite direction).
  4. Click "Calculate Resultant Velocity".
  5. The result will be v: the velocity of the object as observed from the stationary frame S (Earth).

Theory & Formula

The relativistic formula for adding velocities along the same line (collinear velocities) is:

v = (u + v') / (1 + (uv' / c2))

Where:

  • v: The resultant velocity of the object as measured in the stationary frame S. This is what we are calculating.
  • u: The velocity of the moving frame S' relative to the stationary frame S. (Your first input)
  • v': The velocity of the object as measured in the moving frame S'. (Your second input)
  • c: The speed of light in a vacuum (approximately 299,792,458 m/s).

Key Implications:

  • If u and v' are much smaller than c, the denominator (1 + uv'/c2) is very close to 1, and the formula approximates the classical v = u + v'.
  • If either u or v' is c (e.g., adding the velocity of light), the resultant velocity v will also be c. This means the speed of light is constant for all inertial observers, regardless of the motion of the source.
  • It is impossible to accelerate an object with mass to the speed of light, and this formula inherently prevents the resultant velocity from exceeding c if u < c and v' < c.

Calculator Inputs

Enter negative for opposite direction.
Enter negative for opposite direction.
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