Hong Life CalculatorsUnderstanding the length contraction calculator — concept and principle
This calculator applies special relativity to find how much shorter a moving object is measured along its direction of motion. It gives the contracted length, the shortening, the length as a percentage of the proper length and the Lorentz factor. It is used in physics courses and to explore what happens near the speed of light.
How it is calculated
L = L₀ √(1 − v²/c²) = L₀ ÷ γ
γ = 1 ÷ √(1 − v²/c²)
Shortening = L₀ (v/c)² ÷ (1 + √(1 − v²/c²))
- L₀ is the proper length, measured in the object's own rest frame; c = 299,792,458 m/s.
- Speed can be entered as a percentage of c or in other units.
The principle
Special relativity says the speed of light is the same for every observer. To keep it so, observers moving relative to each other must disagree about times and lengths. Measuring a moving object's length means marking both ends at the same moment, and observers disagree about what "the same moment" is. The result is that a moving object is shorter along its motion by the factor γ. Widths across the motion do not change. The shortening formula used here avoids rounding loss at everyday speeds, where the effect is far smaller than an atom.
Worked example
Default inputs: L₀ = 100 m, speed 90% of c.
- v/c = 0.9, so √(1 − 0.81) = √0.19 = 0.43589.
- γ = 1 ÷ 0.43589 = 2.29416.
- L = 100 × 0.43589 = 43.589 m, which is 43.589% of the proper length.
- Shortening = 100 − 43.589 = 56.41 m.
| Proper length at rest (m) | Contracted length (m) |
|---|---|
| 25 | 10.8972 |
| 50 | 21.7945 |
| 75 | 32.6917 |
| 100 | 43.589 |
| 125 | 54.4862 |
| 150 | 65.3835 |
| 200 | 87.178 |
How to read the result
Contraction is negligible at everyday speeds and only becomes large close to c: at 50% of c the length is still about 87%, while at 99% it is about 14%. The proper length is always the longest measured length. Fast muons reaching the ground from the upper atmosphere are a classic test: in their frame, the atmosphere is contracted.
Common mistakes
- Applying contraction to dimensions perpendicular to the motion.
- Mixing up proper length and contracted length.
- Thinking the object is physically compressed; it is a difference in measurement between frames.
Frequently asked questions
Would a moving object look shorter in a photograph?
Not simply; light-travel-time effects make it appear rotated or distorted rather than just contracted.
How fast must something go to halve its length?
About 86.6% of the speed of light, where γ = 2.
Does the moving object notice anything?
No. In its own frame it has its proper length.
How is this linked to time dilation?
Both use the same γ; moving clocks run slow by the same factor that lengths contract.
Sources: OpenStax, University Physics Volume 3 (2016), section 5.4 "Length Contraction"