Relativistic Velocity Addition Calculator
Combine two collinear velocities with Einstein's special-relativity formula, compare the result with ordinary addition, and export the current calculation to a validated Excel workbook.
Inputs
Choose the unknown quantity, then provide the other two velocities.
u = (v + w) / (1 + vw/c²)
Live result
Canonical calculations use the exact speed of light, 299,792,458 m/s.
216,871,139.83 m/s
Signed ratio to c
v + w without relativity
Classical result minus relativistic result
Magnitude of the correction relative to v + w
Sign in the chosen one-dimensional axis
Interpretation snapshot
Distance from +c in the positive direction
1 + vw/c² for the displayed u formula
Based on the largest absolute β
Classical versus relativistic result
| Method | Velocity in selected unit | Fraction of c |
|---|
The relativistic formula gives a lower same-direction speed than ordinary addition while preserving the invariant light-speed limit.
Velocity conversion table
| Quantity | Symbol | Fraction of c | m/s | km/s | mph |
|---|---|---|---|---|---|
| Spaceship velocity | v | 0.350000 | 104,927,360.30 | 104,927.36 | 234,715,820.28 |
| Projectile velocity | w | 0.500000 | 149,896,229.00 | 149,896.23 | 335,308,314.69 |
| Relative projectile velocity | u | 0.723404 | 216,871,139.83 | 216,871.14 | 485,126,923.38 |
All rows come from the same canonical model used by the result cards, comparison chart, and Excel export. Positive and negative signs indicate direction along one chosen axis.
How to use this velocity addition calculator
What this calculator does
This calculator transforms collinear velocities between inertial frames using the one-dimensional special-relativity equation. It finds one of three quantities: Spaceship velocity (v), the moving frame's velocity relative to an outside observer; Projectile velocity (w), the projectile's velocity measured inside that moving frame; or Relative projectile velocity (u), the projectile's velocity measured by the outside observer. It also shows the ordinary Classical sum for comparison. The calculation is an exact algebraic identity within the idealized one-dimensional model; it does not model acceleration, gravity, curved trajectories, propagation through a material, or non-collinear vector components.
When to use it
Use it to check textbook exercises in special relativity, explore how rapidly the Newtonian rule v + w departs from relativistic physics, transform a particle's measured velocity between two collinear inertial frames, or test limiting cases such as one input approaching the speed of light. For perpendicular or oblique motion, use the full vector velocity-transformation equations instead of this one-axis form.
How to calculate
- The calculator opens with a ready-to-use demonstration: v = 0.35 c, w = 0.50 c, and u = 0.723404 c. The validated example workbook is immediately available through Download Excel.
- Choose the desired unknown under Solve for. The selected velocity becomes read-only, while the other two remain editable.
- Select Velocity unit. The choices are fraction of c, m/s, km/s, and mph. Existing finite values are converted rather than merely relabeled.
- Replace the two editable values. The calculator accepts a period as the decimal separator and optional correctly grouped thousands commas. Scientific notation and ambiguous decimal commas are rejected. Each input must have an absolute speed no greater than c.
- Read the primary result, comparison metrics, bar chart, and conversion table. Then use Download Excel to export the current typed inputs and canonical outputs. Reset clears the demonstration values and results; Excel export stays disabled until two complete valid velocities are entered again.
Input guide
- Solve for
- Required select control. Choose u, v, or w as the unknown. Example: choose Relative projectile velocity (u) when v and w are known. Changing it rearranges the same relativistic identity; a common mistake is to edit the read-only target instead of the two source values.
- Velocity unit
- Required unit control shared by all three velocities. Use fraction of c for compact relativistic values, or m/s, km/s, or mph for dimensional values. Example: 0.35 c converts to 104,927,360.3 m/s. Changing the unit does not change the physical speed.
- Spaceship velocity (v)
- Required unless v is selected as the unknown. Enter a signed one-dimensional speed from -c to +c; 0.35 in fraction-of-c mode is a realistic example. Higher positive v usually raises positive u, while a negative value represents the opposite axis direction. Do not enter a percentage such as 35%.
- Projectile velocity (w)
- Required unless w is the unknown. It is measured in the spaceship frame, with the same unit and range rules. The demonstration uses 0.50 c. Its sign is directional, so a negative projectile velocity can partially or fully offset positive v.
- Relative projectile velocity (u)
- Required unless u is the unknown. It is the projectile speed seen by the outside observer. The demonstration result is 0.723404 c. When supplied as an input, it is combined with the other known velocity through the inverse transformation. Inputs that make the inverse denominator zero are physically indeterminate and are rejected.
Output guide
The main Relative projectile velocity, Spaceship velocity, or Projectile velocity result is shown in the selected unit and again in m/s. Fraction of light speed (β) is the signed ratio to c. Classical sum is v + w and can exceed c because it intentionally ignores relativity. Relativistic correction is the classical result minus the relativistic u result for the current v and w, while Reduction vs classical expresses the magnitude of that gap as a percentage of the classical magnitude. Direction reports positive, negative, or stationary. Light-speed margin shows the remaining positive-direction gap to +c; Denominator factor is 1 + vw/c²; and Speed regime classifies the largest absolute β as everyday, noticeable-relativistic, or relativistic. The conversion table columns show each quantity, symbol, fraction of c, m/s, km/s, and mph. The comparison chart's two series, Classical addition and Relativistic addition, use the same current model values as its accessible table.
Worked example
With v = 0.35 c and w = 0.50 c, the numerator is 0.85 c and the denominator is 1 + (0.35 × 0.50) = 1.175. Therefore u/c = 0.85 ÷ 1.175 = 0.7234042553, so the first-open result is 0.723404 c, equal to 216,871,139.83 m/s. Ordinary addition would produce 0.85 c, making the relativistic correction 0.126596 c, or about 14.8936% of the classical value.
Learn more
The derivation and comparison with Galilean addition are explained in the OpenStax section on relativistic velocity transformation. The calculator uses the exact SI value c = 299,792,458 m/s, consistent with the NIST explanation of the defined speed of light.
How the relativistic formula behaves
For same-direction positive velocities, the denominator 1 + vw/c² is greater than one, so the relativistic result is lower than the classical sum. At everyday speeds, vw/c² is so small that the denominator is effectively one and the two methods agree to ordinary measurement precision. As either input approaches c, the correction becomes decisive and the transformed speed approaches c without crossing it.
Signs matter. A negative velocity means motion along the opposite chosen axis, not a negative speed magnitude. When v and w have opposite signs, the numerator can shrink to zero or reverse direction. The one-dimensional equation also admits light-speed limiting cases, but inverse problems can become indeterminate when the supplied frame velocities collapse the denominator to zero. For broader context on the invariance of light speed and relativistic limits, see NASA's special relativity overview.