Physics & Mechanics

Length Contraction Calculator

Calculate the contracted length of an object moving at relativistic speeds relative to an observer. Essential for special relativity.

m
m/s
Contracted Length (L)
86.583

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The Squeezing of Space

Just as velocity warps time, it also warps physical space. Length Contraction (or Lorentz-FitzGerald contraction) is the phenomenon where a moving object appears to shrink in the direction of motion as measured by a stationary observer.

If a spaceship flies past Earth at near light-speed, an observer on Earth will see the spaceship literally squashed flat along its direction of travel.

The Illusion of Rigidity

Length contraction proves that physical dimensions are not absolute properties of an object. The length of a car depends entirely on who is measuring it and how fast they are moving relative to it.

Importantly, length contraction only occurs in the direction of motion. If a spherical spaceship flies past you at 99% the speed of light, it will look like a flattened pancake, not a smaller sphere.

The Formula

L=L01v2c2\begin{aligned} L = L_0 \cdot \sqrt{1 - \frac{v^2}{c^2}} \end{aligned}

Where:
L=
Contracted Length (Observed by stationary observer)
L0L_0=
Proper Length (Measured at rest)
v=
Velocity of the object
c=
Speed of Light

Example Calculation

A spaceship is exactly $100 , \text{meters}$ long when parked (Proper Length, $L_0$). It flies past a space station at 86.6% the speed of light.

  1. Calculate the Lorentz Factor denominator: $\sqrt{1 - 0.866^2} \approx \sqrt{0.25} = 0.5$.
  2. Multiply: $100 \times 0.5 = 50 , \text{meters}$.

To the people on the space station, the ship is only $50 , \text{meters}$ long. However, to the pilot inside, the ship is still $100 , \text{meters}$ long.

Frequently Asked Questions

No. There are no physical forces crushing the ship. It is the very fabric of space itself that is contracting relative to the observer. The atoms in the ship don't feel squashed; their entire frame of reference has shrunk.

Technically, yes. If you sprint at $10 , \text{m/s}$, an observer standing still will measure you as being about one quadrillionth of a millimeter thinner. At everyday speeds, the contraction is far too small to measure.

Relativity is symmetrical. To the pilot, the spaceship is perfectly normal, but the space station and the entire universe flying past the window will appear severely length-contracted (squashed).