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, 2) / pow(2.998e8, 2))

Length Reduction (%)

contraction_pct = (1 - sqrt(1 - pow(velocity, 2) / pow(2.998e8, 2))) * 100

Variables

VariableDescriptionDefault
proper_lengthProper Length (L0)(m)100
velocityVelocity(m/s)269800000

How It Works

Lorentz Length Contraction

Objects moving at relativistic speeds appear shorter in the direction of motion.

Formula

L = L0 * sqrt(1 - v^2/c^2) = L0 / gamma

  • *L0* = proper length (measured at rest)
  • *v* = relative velocity

    • Contraction occurs only along the direction of motion. Transverse dimensions are unaffected.

    Worked Example

    A 100 m spaceship at 0.9c.

    proper_length = 100velocity = 269800000
    1. 01beta = 0.9, gamma = 2.294
    2. 02L = L0 / gamma = 100 / 2.294
    3. 03L = 43.59 m
    4. 04The ship appears 56.4% shorter.

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    Open Length Contraction Calculator