Velocity Addition Calculator
This velocity addition calculator will help you better understand special relativity and the relativistic velocity addition formula. You might have heard that velocities cannot simply be added together. Instead, we should use the Einstein velocity addition equation, sometimes also called the relativistic velocity addition equation.
Interested in relativity? Check out the relativistic kinetic energy calculator or the electron speed calculator!
The Speed of Light is the Same for Everyone
Imagine your friend flying a spaceship at high speed and firing a projectile. What is the speed of the projectile? If the spaceship's speed is v and your friend fired the projectile at speed w (with respect to the spaceship), then it seems logical that you will see the projectile traveling at speed v + w.
However, this cannot be true. If your friend turns on a laser instead of firing a projectile, then the speed of light he observes is, well, the speed of light c. The speed of light is the same for every observer, which means that you will also see the speed of light as c, not v + c. This argument shows that we cannot just add velocities.
This is where our relativistic velocity addition calculator can help resolve this contradiction. Keep reading to find out more.
Time and Space in Special Relativity
Both you and your friend observe the same speed of light because, when moving, time passes at different rates and distances appear different. These two effects are time dilation and length contraction. You can see how they work using the time dilation calculator and the length contraction calculator. If we combine these two effects, the result is the relativistic velocity addition formula.
Relativistic Velocity Addition Formula
How do we add the velocities? The formula is:
u = (v + w) / (1 + vw/c²)
where:
- u — Speed of the projectile as seen from outside the spaceship;
- v — Speed of the spaceship;
- w — Speed of the projectile as seen from the spaceship; and
- c — Speed of light (299,792,458 m/s).
When the speed of the spaceship or the projectile's speed is slow compared to the speed of light, the formula reduces to the simple sum of the velocities u = v + w.
However, if the velocities v and w are both large, then the projectile's speed, as seen from outside the spaceship, is much lower than their sum v + w and never exceeds the speed of light. In the extreme case, when one of the velocities equals the speed of light c, the other one won't exceed it either — the speed of light is the same for everyone.
You can check other effects of special relativity with the famous E = mc² calculator.
Measurement Systems
This calculator supports both the metric system (m/s, km/h) and the American (imperial) system (mph, ft/s), as well as entering each velocity as a fraction of the speed of light (e.g. 0.8 means 0.8c = 80% of c). The results include a full velocity conversion table so you can read the answer in whichever units you prefer.
How to Use This Calculator
- Enter the spaceship velocity (v) and choose its unit.
- Enter the projectile velocity (w), measured relative to the spaceship, and choose its unit.
- Click Calculate. The calculator shows the relativistic sum u, compares it with the naive classical sum v + w, and lists the result in m/s, km/h, mph, ft/s, and as a fraction of c.
Frequently Asked Questions
Why can't I just add the two velocities?
At everyday speeds you essentially can — the relativistic correction is far too small to notice. But as the velocities approach the speed of light, simply adding them would give a result greater than c, which is forbidden by relativity. The Einstein velocity addition formula always keeps the result below (or equal to) c.
What happens if I add c to any velocity?
If either input equals the speed of light, the result is exactly c. Light travels at c for every observer, regardless of the motion of its source.
Is the order of v and w important?
No. The relativistic velocity addition formula is symmetric in v and w, so swapping them gives the same result.
What is a "fraction of c"?
When you enter a velocity as a fraction of c (e.g. 0.6), you are saying the object moves at 0.6 × 299,792,458 m/s ≈ 179,875,475 m/s. This notation is common in physics because relativistic effects only become significant when the velocity is a substantial fraction of c.