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Length Contraction Calculator

Find how length contracts at relativistic speeds.

How it is calculated

The tool applies special-relativity length contraction: L = L₀ × √(1 − β²), where L₀ is the proper length in metres (measured in the object's own rest frame) and β = v ÷ c is the speed as a fraction of the speed of light, from 0 up to but not including 1. It also shows the Lorentz factor γ = 1 ÷ √(1 − β²), so L = L₀ ÷ γ. Only the length along the direction of motion shrinks; width and height are unchanged. Speed must be entered as a fraction of c, not in km/h or m/s. A value of 1 or more gives no result, because an object with mass cannot reach light speed. Results show four decimals.

Worked example

A 10 m rod moving at β = 0.8. β² = 0.64, so 1 − β² = 0.36 and √0.36 = 0.6. Contracted length: 10 × 0.6 = 6.0000 m. Lorentz factor: 1 ÷ 0.6 = 1.6667. An observer who sees the rod fly past measures it as 6 m long, while someone riding with it still measures 10 m.

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Common mistakes

  • Entering the speed in km/h or m/s. The field wants v ÷ c, so 0.8 means 80% of light speed; 240000 gives no result.
  • Entering 80 for 80%. The input is a fraction between 0 and 1, so 80% is 0.8.
  • Expecting the contraction to appear at everyday speeds. At β = 0.001 the length changes by only about five parts in ten million.
  • Thinking the object itself is physically squashed. The shortening is what an observer in a different frame measures; in its own frame the object keeps its proper length.
  • Applying it to width or height. Only the length along the direction of motion contracts.

About this calculator

Length contraction is a consequence of Einstein's special relativity: an object moving at velocity v relative to an observer appears shortened along its direction of motion, following L = L₀·√(1 − β²), where L₀ is the object's rest-frame (proper) length, β = v/c is its speed as a fraction of the speed of light, and the factor √(1 − β²) is the inverse of the Lorentz factor γ. The effect is negligible at everyday speeds but becomes dramatic as v approaches c — at 86.6% of light speed an object appears contracted to half its rest length.

Physics students use this calculator to build intuition for special relativity problems alongside time dilation, particle physicists apply the same Lorentz factor when analyzing particles accelerated to near-light speed in colliders, and science communicators and writers use it to check the numbers behind relativistic scenarios in popular explanations or fiction.

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Comments (1)

  • Editorial team

    To use a real speed, divide it by the speed of light first: about 299,792 km/s. Earth orbits the Sun at roughly 30 km/s, so beta is 0.0001, and a 10 m rod still shows 10.0000 m because the change is far below the fourth decimal. At beta 0.99, by contrast, a 100 m ship measures about 14.1 m. Are you checking a physics exercise or the numbers behind a story?

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