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Speed as a Fraction of c Calculator

Enter a speed to see what fraction of the speed of light it is.

How it is calculated

The speed is first converted to metres per second: km/s is multiplied by 1000, and km/h by 1000/3600. Then β = v / c with c = 299,792,458 m/s, the percentage is β × 100, and the Lorentz factor is γ = 1 / √(1 − β²). β and γ are shown to six decimals and the percentage to two. The speed must be above zero and strictly below c, so zero, negative values and light speed itself return no result. Only the units m/s, km/s and km/h are offered. At everyday speeds β is so small that it shows as 0.000000 and γ as 1.000000, so the tool is only informative for very fast objects.

Worked example

Take 60,000 km/s. That is 60,000,000 m/s. Dividing by 299,792,458 gives β = 0.200138, so the speed is 20.01% of the speed of light. Then β² ≈ 0.040055, 1 − β² ≈ 0.959945, and its square root is about 0.979768. So γ = 1 / 0.979768 ≈ 1.020650. At this speed clocks run about 2% slower than at rest in the observer's frame, a small but measurable relativistic effect.

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Common mistakes

  • Choosing the wrong unit. 60,000 read as m/s is a tiny fraction of c, but as km/s it is 20% of c. Check the unit selector before reading the result.
  • Expecting the percentage to show small values precisely. It has two decimals, so a very small percentage appears as 0.00%. Use β for very small speeds, and expect even that to show 0.000000 for car or aircraft speeds.
  • Reading γ as the speed ratio. γ is the time-dilation factor and is always at least 1, whereas β is between 0 and 1.
  • Entering light speed or more. The tool requires v below c, so 299,792,458 m/s returns no result.
  • Using miles per hour. Only m/s, km/s and km/h can be selected, so convert mph yourself first (1 mph is about 1.609 km/h).

About this calculator

In special relativity, speeds are naturally expressed relative to the speed of light, c ≈ 299,792,458 m/s, using the dimensionless ratio β = v/c. This calculator takes any speed and reports β, its equivalent percentage of c, and the Lorentz factor γ = 1/√(1 − β²), the multiplier that governs how much time dilation, length contraction, and relativistic mass increase occur at that speed.

Physicists and astronomers use β and γ constantly: particle accelerators quote beam energies in terms of γ, astrophysicists describe jets from black holes or gamma-ray bursts as “relativistic” when β approaches 1, and GPS satellite clock corrections rely on the same γ factor at far smaller speeds. Because γ grows without bound as v approaches c, this tool makes it easy to see just how close to light speed a given value actually is — for everyday speeds β is vanishingly small and γ is indistinguishable from 1, while for particle-accelerator speeds γ can reach into the thousands.

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