Velocity Addition Calculator

In classical physics, velocities simply add together linearly. However, according to special relativity, velocities combine non-linearly to ensure that no object can ever exceed the speed of light in a vacuum ($c \approx 299,792,458\text{ m/s}$). The formula used by this calculator is expressed as:

$v = \frac{u + v'}{1 + \frac{u v'}{c^2}}$

Where $v$ is the resultant relative velocity, $u$ is the velocity of the reference frame (such as a spaceship), $v'$ is the velocity of the projectile relative to that reference frame, and $c$ is the speed of light.

Imagine a futuristic starship traveling at $0.6c$ (60% the speed of light) relative to Earth. Inside this starship, a crew fires a probe forward at a velocity of $0.5c$ relative to the starship. To find the probe's speed relative to Earth, we input $u = 0.6c$ and $v' = 0.5c$ into the relativistic equation.

1. Multiply the two velocities: $(0.6c) \times (0.5c) = 0.3c^2$.
2. Divide by $c^2$ to get $0.3$.
3. Add $1$ to the denominator: $1 + 0.3 = 1.3$.
4. Add the numerator velocities: $0.6c + 0.5c = 1.1c$.
5. Divide the numerator by the denominator: $1.1c / 1.3 \approx 0.846c$.

Thus, the probe travels at approximately $84.6\%$ the speed of light relative to Earth, perfectly obeying the cosmic speed limit rather than reaching the naive classical sum of $1.1c$.

When performing high-speed calculations, always ensure your units are consistent—typically meters per second ($ ext{m/s}$) or expressed as a fraction of the speed of light ($c$). Keep in mind that for everyday speeds (like cars or commercial jets), the denominator approaches $1$, making the relativistic formula yield nearly identical results to classical Galilean addition. Use this calculator specifically when speeds exceed roughly $10\%$ of the speed of light ($0.1c$), where relativistic effects become genuinely measurable.