Length Contraction Calculator
length contraction calculator. Get instant, accurate results. Enter values for instant results with step-by-step formulas.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
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Formula: L = L₀/γ = L₀√(1-v²/c²)
Worked example — 60 m (40% shorter)
L = L₀/γ = L₀√(1-v²/c²)
Objects moving at relativistic speeds appear shorter in the direction of motion.
Problem:L₀=100m, v=0.8c
Solution:γ=1.667, L=100/1.667=60m
Result:60 m (40% shorter)
Length contraction is the phenomenon where an object moving at relativistic speed appears shorter along its direction of motion to a stationary observer. The contracted length is L = L₀/γ = L₀√(1 − v²/c²), where L₀ is the proper (rest-frame) length.
First compute β = v/c, then γ = 1/√(1 − β²). Finally L = L₀/γ. Example: a 100 m spaceship at v = 0.8c gives γ = 1/√(1−0.64) = 1.667, so L = 100/1.667 ≈ 60 m.
Proper length L₀ and contracted length L are both in metres (m). Velocity v can be entered in m/s or as a fraction of c (e.g. 0.8c). The speed of light c = 2.998×10⁸ m/s. γ (Lorentz factor) is dimensionless.
Yes. Atmospheric muons created at ~15 km altitude survive to Earth's surface because in their rest frame the atmosphere is contracted to ~0.5 km. Particle accelerators also routinely rely on relativistic length and time effects for accurate beam design.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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