Relativistic Velocity Addition Calculator

Relativistic velocity addition: the Einstein formula

In everyday life we simply add velocities: if you walk at 3 m/s on a train moving at 20 m/s, someone on the ground says you move at 23 m/s. This is the Galilean or classical velocity-addition rule.

At speeds close to the speed of light \(c\), this rule breaks down. Using \(u + v\) can give speeds greater than \(c\), which is forbidden in special relativity. Instead we must use Einstein’s relativistic velocity-addition formula.

1D relativistic velocity addition formula

Consider two inertial frames:

• Frame \(S\):\) the “lab” frame.
• Frame \(S'\):\) moving at speed \(v\) along the +x direction relative to \(S\).

An object moves at speed \(u\) along +x in frame \(S\). What is its speed \(w\) in frame \(S'\)? Special relativity gives:

Here \(u\), \(v\), and \(w\) are signed velocities along the same line (positive or negative). No matter how large \(u\) and \(v\) are (as long as \(|u| < c\) and \(|v| < c\)), the result always satisfies \(|w| < c\).

2D velocity transformation (perpendicular component)

If the object has components \((u_x, u_y)\) in frame \(S\), and \(S'\) moves at speed \(v\) along +x, the velocity components in \(S'\) are:

Our 2D mode uses these formulas to compute the transformed components, the resulting speed \(|\mathbf{u}'|\), and the direction angle in the moving frame.

Lorentz factor and rapidity

Two useful quantities in special relativity are the Lorentz factor \(\gamma\) and the rapidity \(\eta\).

• Lorentz factor: \[ \gamma(v) = \frac{1}{\sqrt{1 - v^2/c^2}} \] It appears in time dilation, length contraction, and energy–momentum relations.
• Rapidity: \[ v = c \tanh(\eta), \qquad \eta = \operatorname{artanh}\left(\frac{v}{c}\right) \] In 1D, rapidities add linearly: \(\eta_{\text{total}} = \eta_1 + \eta_2\). Our calculator shows \(\eta(u)\), \(\eta(v)\), and \(\eta(w)\) so you can verify this.

How to use the relativistic velocity addition calculator

1. Select Mode:
   • 1D for collinear velocities (most textbook problems).
   • 2D when the object has a perpendicular component (e.g. a particle moving at an angle).
2. Choose Velocity units:
   • Fraction of c (e.g. 0.8 means \(0.8c\)).
   • m/s or km/h for everyday units (the tool converts internally).
3. Enter the velocities:
   • 1D: enter \(u\) and \(v\).
   • 2D: enter \(u_x\), \(u_y\), and \(v\) (frame speed along x).
4. Click Compute relativistic sum.
5. Compare the relativistic result with the classical sum and inspect the gamma factors and rapidities.

Worked examples

Example 1: Two spaceships approaching head-on

Spaceship A moves at \(+0.8c\) relative to Earth, and spaceship B moves at \(-0.8c\) relative to Earth (opposite directions). What is the speed of B as seen from A?

In 1D mode with units as fraction of c:

• Set \(u = 0.8c\) (A relative to Earth).
• Set \(v = -0.8c\) (B relative to Earth, opposite direction).

Relativistic relative speed:

Classically you might say the ships approach at \(1.6c\), but relativity gives about \(0.976c\), safely below the speed of light.

Example 2: Perpendicular component in 2D

A particle moves in frame \(S\) with components \(u_x = 0.6c\), \(u_y = 0.8c\). Frame \(S'\) moves at \(v = 0.5c\) along +x relative to \(S\). What is the particle’s velocity in \(S'\)?

In 2D mode, fraction-of-c units:

• Enter \(u_x = 0.6\), \(u_y = 0.8\), \(v = 0.5\).

The calculator will show:

• Relativistic components \(u'_x\) and \(u'_y\).
• Resulting speed \(|\mathbf{u}'|\) and direction angle in \(S'\).
• Classical components for comparison, where you would simply subtract \(v\) from the x-component and leave y unchanged.

Common pitfalls and tips

• Never exceed c: If your classical sum exceeds \(c\), that’s a sign you must use the relativistic formula.
• Sign matters: Use negative velocities for motion in the opposite direction along the chosen axis.
• Use fractions of c for clarity: For relativistic problems, it’s often easiest to think in units of \(c\) (e.g. 0.3c, 0.9c).
• Check the denominator: The factor \(1 + uv/c^2\) (or \(1 - u_x v/c^2\) in 2D) is what keeps the result below \(c\).