Length Contraction Calculator
Free calculate length contraction in special relativity from proper length and velocity. Get instant, accurate results with our easy-to-use calculator.
Input Parameters
Length measured in object's rest frame
Relative velocity between observer and object
Results
Enter parameters to calculate
What is Length Contraction?
Length contraction is a phenomenon in special relativity where an object moving relative to an observer appears shorter along the direction of motion than when measured in its rest frame. This effect becomes significant at velocities approaching the speed of light.
The proper length (L₀) is the length measured in the object's rest frame. To an observer moving relative to the object, the length appears contracted. This is a real physical effect, not an optical illusion, and is a consequence of the constancy of the speed of light.
Length contraction is one of the key predictions of Einstein's special theory of relativity, along with time dilation. It has been experimentally verified in particle accelerators and is essential for understanding high-speed physics.
Length Contraction Formula
Where:
- • L = Contracted length (observed length)
- • L₀ = Proper length (rest frame length)
- • v = Relative velocity
- • c = Speed of light (299,792,458 m/s)
- • γ = Lorentz factor = 1/√(1 - v²/c²)
Note: Length contraction only occurs along the direction of motion. Perpendicular dimensions are unchanged.
How to Calculate
-
1
Convert velocity to m/s
If given in units of c, multiply by c = 299,792,458 m/s. Convert km/h to m/s (divide by 3.6).
-
2
Calculate Lorentz factor
γ = 1/√(1 - v²/c²). This factor quantifies the relativistic effects.
-
3
Calculate contracted length
L = L₀ / γ = L₀ × √(1 - v²/c²)
Practical Examples
Example 1: High-Speed Spacecraft
A 100 m spacecraft travels at 0.8c. Calculate contracted length.
Solution:
γ = 1/√(1 - 0.8²) = 1/√(0.36) = 1/0.6 = 1.667
L = 100 m / 1.667 = 60 m
The spacecraft appears 40% shorter to a stationary observer!
Example 2: Relativistic Particle
A muon with proper lifetime travels at 0.99c. What is the contraction?
Solution:
γ = 1/√(1 - 0.99²) = 1/√(0.0199) ≈ 7.09
Length contracts to about 14% of proper length at 0.99c
Applications
Particle Physics
Understanding particle lifetimes, accelerator physics, and high-energy collisions where particles move near light speed.
Astrophysics
Analyzing relativistic jets, cosmic rays, and high-speed astronomical phenomena where relativistic effects are significant.
GPS Systems
Accounting for relativistic effects in satellite clocks and positioning systems, where precision requires relativistic corrections.
Education
Teaching special relativity, understanding space-time, and demonstrating counterintuitive relativistic effects.
Frequently Asked Questions
Is length contraction real or just an optical effect?
Length contraction is a real physical effect, not an optical illusion. It's a consequence of the constancy of the speed of light and has been experimentally verified in particle accelerators.
Why don't we notice length contraction in everyday life?
At everyday speeds (much less than c), the contraction factor √(1 - v²/c²) is extremely close to 1, making the effect negligible. At 100 km/h, contraction is only about 0.000000000004%!
Does the object actually get shorter?
From the observer's frame, yes - the length is genuinely shorter. However, from the object's own frame, its length is unchanged. This is the relativity of simultaneity - different observers measure different lengths.
What happens at the speed of light?
As v approaches c, the length approaches zero (L → 0). However, objects with mass cannot reach the speed of light, so this is a theoretical limit. Massless particles (photons) always travel at c.
How is length contraction related to time dilation?
Both are consequences of special relativity and the Lorentz factor γ. Time dilation: Δt = γ×Δt₀. Length contraction: L = L₀/γ. They're complementary effects - time stretches while length contracts.