Length Contraction Calculator Formula
Understand the math behind the length contraction calculator. Each variable explained with a worked example.
Formulas Used
Contracted Length
contracted_length = proper_length * sqrt(1 - pow(velocity_fraction, 2))Contraction
contraction_pct = (1 - sqrt(1 - pow(velocity_fraction, 2))) * 100Variables
| Variable | Description | Default |
|---|---|---|
proper_length | Proper Length (rest frame)(m) | 100 |
velocity_fraction | Velocity (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.
- 01L = L0 * sqrt(1 - v²/c²)
- 02L = 100 * sqrt(1 - 0.81)
- 03L = 100 * sqrt(0.19)
- 04L = 100 * 0.4359
- 05L = 43.59 m (56.4% shorter)
Frequently Asked Questions
Does the object actually shrink?
The object does not physically compress. Length contraction is a consequence of the geometry of spacetime. In its own reference frame, the object retains its proper length.
At what speed is contraction noticeable?
The effect is negligible below about 10% of c. At 86.6% of c (gamma=2), length is halved. At 99.5% of c, length contracts to 10%.
Is length contraction related to time dilation?
Yes. Both are consequences of the Lorentz transformation. Length contracts while time dilates, keeping the speed of light constant in all frames.
Ready to run the numbers?
Open Length Contraction Calculator