Length Contraction Calculator
Calculate relativistic length contraction from proper length and speed, entered either in m/s or as a fraction of the speed of light.
Length Contraction Calculator
Calculator
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Formula: L = L₀/γ = L₀√(1-v²/c²)
Worked example — 60 m (40% shorter)
Formula
L = L₀/γ = L₀√(1-v²/c²)
Objects moving at relativistic speeds appear shorter in the direction of motion.
Worked Examples
Example 1: Spaceship at 0.8c
Problem:L₀=100m, v=0.8c
Solution:γ=1.667, L=100/1.667=60m
Result:60 m (40% shorter)
Frequently Asked Questions
What is length contraction in special relativity?
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.
How do you calculate the Lorentz factor γ and contracted 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.
What units are used for length contraction calculations?
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.
Is length contraction experimentally confirmed?
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.
What is the difference between length contraction and time dilation?
Both are governed by the same Lorentz factor gamma but act in opposite senses: lengths shrink to L_0/gamma, while time intervals stretch to gamma x delta-t_0. They are two views of one geometry. For atmospheric muons, an Earth-based observer says the muon's clock runs slow so it lives long enough to arrive; the muon says the atmosphere is contracted so there is less distance to cross. Both frames predict the same outcome.
Does an object contract in every direction?
No - only along the direction of relative motion. Dimensions perpendicular to the velocity are completely unaffected. A sphere moving at 0.8c is measured as an ellipsoid squashed to 0.6 of its rest diameter along the motion while keeping its full width across it. That is why the calculator asks only for the length parallel to the velocity, and why the transverse cross-section stays unchanged.
Which observer measures the proper length?
Proper length L_0 is measured in the frame where the object sits at rest, and it is always the largest value any observer will get. Every frame in relative motion measures something shorter. The subtlety is that measuring a moving object's length requires noting both ends at the same instant in your own frame, and simultaneity itself is frame-dependent - that relativity of simultaneity is the deeper reason contraction occurs.
At what speeds does length contraction become noticeable?
It is negligible until velocities approach a substantial fraction of c because gamma grows slowly at first. At 0.1c, gamma = 1.005 and the contraction is only 0.5 percent. At 0.5c, gamma = 1.155 and the object measures 0.866 of its rest length, a 13.4 percent reduction. At 0.9c, gamma = 2.294 and it measures just 0.436 of its rest length - a 56.4 percent contraction.
Would a fast-moving object actually look contracted in a photograph?
Not simply, no. What a camera or eye records depends on when light from each part of the object left it, and light from the far side has further to travel. The combined effect, known as Terrell-Penrose rotation, makes a passing object appear rotated or sheared rather than merely compressed. The contraction Length Contraction Calculator returns is the measured length using simultaneous readings, not the photographed shape.
Why can nothing with mass reach the speed of light?
As beta approaches 1, gamma diverges, so the contracted length would fall to zero and the energy required, gamma x m x c-squared, would become infinite. No finite input can push a massive object all the way to c; it can only be brought asymptotically closer. That is why the calculator rejects any velocity at or above c, where the square root of (1 - beta-squared) becomes zero or imaginary.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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