Barn Pole Paradox Calculator

The Barn-Pole Paradox

A runner carries a pole longer than a barn and tries to fit it inside by closing both doors while fully inside. Special relativity creates a paradox: in the barn frame the pole contracts and fits, but in the pole frame the barn contracts and is too short. The resolution involves simultaneity and the finite speed of door closure.

Lorentz Contraction

$$L' = \frac{L}{\gamma}, \quad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$$

Lengths parallel to motion contract in the observer's frame. At $v = 0.866c$, $\gamma = 2$. A 13 m pole contracts to 6.5 m while a 10 m barn stays 10 m in the barn frame, so the pole appears to fit momentarily.

Frame-by-Frame Comparison

In the barn frame, the moving pole is shortened. In the pole frame, the barn is shortened. Both are correct locally; the paradox arises only if you assume absolute simultaneity. This calculator shows contracted lengths and whether each frame predicts a fit at the instant of overlap.

Frequently Asked Questions

Who is right, the barn observer or the runner?

Both measurements of contracted length are valid in their frames. The full story requires spacetime diagrams and careful timing of door events.

What velocity gives γ = 2?

$v = \sqrt{3}/2 \approx 0.866c$. At this speed, lengths contract by half.

Does the pole actually break?

Classical rigidity cannot exist at near-light speeds. Real materials transmit forces at finite speed, so the thought experiment highlights relativity, not engineering.

Why use meters for both lengths?

Any common unit works because only ratios matter. Meters keep the example aligned with textbook values (10 m barn, 13 m pole).

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The Barn-Pole Paradox

Lorentz Contraction

Frame-by-Frame Comparison

Frequently Asked Questions

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