The relativistic velocity addition calculator above answers a question that classical mechanics gets wrong in an interesting way. If a ship moves at half the speed of light past you and fires a probe forward at half the speed of light relative to itself, how fast is the probe going in your frame? Adding gives c. The correct answer is 0.8c, and the difference is not a rounding error — it is the whole content of the formula.
Arb Digital publishes free physics calculators that state where their formula applies and where it stops. The classical sum u′ + v is not wrong so much as it is a limit: it is what this formula becomes when both speeds are small compared with c. This page shows both numbers side by side so the size of the correction is always visible, and it exists mainly to make one point concrete — no combination of subluminal velocities ever produces a result at or above c.
What This Relativistic Velocity Addition Calculator Does
You give it two velocities along the same line. The first, u′, is measured in a moving reference frame. The second, v, is the velocity of that frame relative to you. The calculator returns u, the velocity measured in your frame, using the Lorentz velocity transformation.
Alongside it you get the classical sum, the shortfall between the two, the Lorentz factor of the result, and the percentage error you would have made by adding naively. That last figure is the useful one for deciding whether relativity matters in a given problem: at aircraft speeds it is around 10−10 per cent, and at 0.5c it is 25 per cent.
Negative values are accepted and mean motion in the opposite direction, which is how you handle two objects approaching each other head-on. That case is worth running, because the relative speed of two objects each moving at 0.9c towards one another is not 1.8c but about 0.9945c.
How to Use It
- Enter u′, the velocity measured inside the moving frame. A fraction of c is usually the clearest unit for this.
- Enter v, the velocity of that frame relative to you. The formula is symmetric in the two, so the order does not affect the answer.
- Use a negative sign for opposing directions. Head-on approach means one of the two is negative.
- Compare the two grid figures. The classical sum and the shortfall together show exactly how much relativity is contributing.
- Try setting one input to exactly c. The result comes back as c regardless of the other value, which is the postulate the whole theory is built on.
The Formula: How Relativistic Velocities Combine
The relativistic velocity transformation for motion along a common axis is u = (u′ + v) ÷ (1 + u′v ÷ c²). OpenStax University Physics, Volume 3, section 5.6 on relativistic velocity transformation states it in exactly this form and works the case of a spaceship heading towards Earth at half the speed of light emitting a laser beam, showing that the light still arrives at c rather than at 1.5c.
The denominator is the entire difference from classical mechanics. When u′v is small compared with c², that term is close to 1 and the expression collapses to the ordinary sum. When both velocities approach c, the product approaches c², the denominator approaches 2, and the numerator's 2c is halved back to c.
These relations follow from applying the Lorentz transformation to velocities rather than positions. The Einstein velocity addition page at Georgia State University's HyperPhysics makes the resulting guarantee explicit: the relative velocity of any two objects never exceeds the velocity of light, and that is a property of the transformation itself rather than an extra assumption bolted on.
Work the defaults by hand. With u′ = 0.5c and v = 0.5c, the numerator is 1.0c and the denominator is 1 + (0.5)(0.5) = 1.25, so u = 1.0c ÷ 1.25 = 0.8c. In SI that is 0.8 × 299,792,458 = 239,833,966.4 m/s. The classical sum would have been 299,792,458 m/s, so the shortfall is 0.2c and the classical answer is 25 per cent too high relative to the correct one. The Lorentz factor at 0.8c is 1 ÷ √(1 − 0.64) = 1 ÷ 0.6 = 1.6667.
Why the Answer Can Never Reach c
This is the point of the page, so it is worth showing rather than asserting. Suppose both u′ and v are strictly less than c. Write them as fractions α and β, both below 1. Then u÷c = (α + β) ÷ (1 + αβ). For this to equal 1 you would need α + β = 1 + αβ, which rearranges to (1 − α)(1 − β) = 0. That requires one of them to be exactly 1 — exactly c.
So as long as both inputs are genuinely below c, the output is genuinely below c, no matter how close either one gets. Combine 0.99c with 0.99c and you get 0.99995c. Combine that with 0.99c again and you get 0.9999997c. The result creeps towards the limit and never arrives.
The converse case is just as sharp. Set either input to exactly c and the output is exactly c whatever the other input is: (c + v) ÷ (1 + v÷c) = c(c + v) ÷ (c + v) = c. Light travels at the same speed in every inertial frame, and here that is not a postulate being imposed but an algebraic consequence of the transformation.
The Classical Sum as a Low-Speed Limit
It is tempting to think of Galilean addition as simply wrong. It is more accurate, and more useful, to think of it as the first term of an expansion. Expanding the denominator for small u′v÷c² gives u ≈ (u′ + v)(1 − u′v÷c²), so the leading correction is of order u′v÷c².
Put ordinary numbers in and the size of that correction becomes obvious. Two aircraft at 300 m/s give u′v÷c² of about 10−12, which is a correction in the twelfth decimal place. No aircraft instrument in existence could detect it. Measured against the correct answer, the classical sum is about 1 per cent high when two 0.1c velocities are combined, 25 per cent high at 0.5c and 81 per cent high at 0.9c.
This is why classical mechanics survived unchallenged for two centuries. It is not that anyone was careless; it is that the correction is genuinely invisible at every speed people could produce or measure until the twentieth century. The same pattern appears in the relativistic kinetic energy calculator, which shows the classical one-half m v squared alongside the correct expression and the percentage error between them.
Where the Formula Applies and Where It Does Not
Everything above is the collinear case: both velocities along the same straight line. When the velocity in the moving frame has a component perpendicular to the boost, that component transforms too, and not simply by staying the same. It is divided by both the Lorentz factor and the same denominator, which is why a light beam emitted sideways from a moving source arrives at an angle — the aberration of starlight, measured long before relativity explained it.
The transformation also applies only between inertial frames. Accelerating frames need the fuller machinery of relativistic kinematics, and in curved spacetime the notion of the relative velocity of two distant objects stops being well defined at all. That is why cosmological recession velocities can exceed c without contradicting anything here: they are not velocities in a single inertial frame.
Within its domain the formula is exact, not approximate. It has been confirmed directly, most famously by measuring the speed of light emitted from fast-moving particle beams, and indirectly by every particle accelerator that works. For the other consequences of the same transformation, see the time dilation calculator and the length contraction calculator, and for the gravitational case the gravitational time dilation calculator.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Adding the velocities directly — the denominator is the whole physics. Leaving it out gives answers above c, which never occur.
- Mixing frames — u′ must be measured in the moving frame and v must be that frame's velocity relative to you. Two velocities both measured in your frame do not combine with this formula.
- Using it for perpendicular motion — the transverse component transforms differently, picking up a factor of the Lorentz factor as well.
- Thinking a closing speed above c is forbidden — in one frame, two objects approaching can have separation shrinking faster than c. What can never exceed c is the speed of one measured in the rest frame of the other.
- Treating classical addition as a competing theory — it is the low-speed limit of this expression, and the calculator shows exactly how far apart they are.
Related Free Tools From Arb Digital
The relativistic kinetic energy calculator gives energy, momentum and the Lorentz factor for a single particle. The time dilation calculator and the length contraction calculator cover the other two standard consequences of the Lorentz transformation, and the gravitational time dilation calculator the general-relativistic case. For the classical baseline use the kinetic energy calculator and the momentum calculator, and for unit work the speed converter. The Schwarzschild radius calculator covers where spacetime curvature becomes extreme. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
It is u equals u prime plus v, all divided by one plus u prime times v over c squared. The denominator is what distinguishes it from ordinary addition, and it is what keeps the result below the speed of light.
You get 0.8c, not c. The numerator is 1.0c and the denominator is 1.25, so the result is four fifths of the speed of light. The classical sum overshoots by 25 per cent relative to the correct answer.
No. Setting the result equal to c requires one of the two inputs to be exactly c, which the algebra shows directly. As long as both are strictly below c, the result is strictly below c no matter how close either one gets.
It is the low-speed limit of the relativistic expression. The leading correction is of order u prime times v over c squared, which for two aircraft is around one part in a trillion. It only becomes visible when speeds are a substantial fraction of c.
The result is exactly c, whatever the other velocity is. That falls straight out of the algebra, and it is the mathematical expression of light travelling at the same speed in every inertial frame.
Enter one velocity as negative. Two objects each moving at 0.9c towards one another have a relative speed of about 0.9945c as measured by either of them, not 1.8c.
Not as written. This is the collinear case. A velocity component perpendicular to the boost direction transforms with an extra factor of the Lorentz factor, which is what produces the aberration of light from a moving source.
This tool is provided for educational and study use. It applies the special-relativistic velocity transformation for collinear motion between inertial frames, and does not cover transverse components, accelerating frames or curved spacetime. Treat its output as a physics result rather than a measured value.
