Length Contraction Calculator Formula

Understand the math behind the length contraction calculator. Each variable explained with a worked example.

Use the Length Contraction Calculator

Formulas Used

Contracted Length

contracted_length = proper_length * sqrt(1 - pow(velocity_fraction, 2))

Contraction

contraction_pct = (1 - sqrt(1 - pow(velocity_fraction, 2))) * 100

Variables

VariableDescriptionDefault
proper_lengthProper Length (rest frame)(m)100
velocity_fractionVelocity (fraction of c)0.9

How It Works

Length Contraction

Objects in relative motion appear contracted along the direction of motion.

Formula

L = L_0 * sqrt(1 - v²/c²) = L_0 / gamma

where L_0 is the proper length (measured at rest). This is only along the direction of motion; perpendicular dimensions are unchanged.

Worked Example

A 100 m spaceship traveling at 90% the speed of light.

proper_length = 100velocity_fraction = 0.9
  1. 01L = L0 * sqrt(1 - v²/c²)
  2. 02L = 100 * sqrt(1 - 0.81)
  3. 03L = 100 * sqrt(0.19)
  4. 04L = 100 * 0.4359
  5. 05L = 43.59 m (56.4% shorter)

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