This barn pole paradox calculator, also called the ladder paradox, shows how special relativity resolves a famous puzzle: a pole that is longer than a barn appears to fit inside it when it moves fast enough. You enter the rest length of the pole, the rest length of the barn, and the speed as a fraction of the speed of light, and the calculator applies length contraction to work out what each observer measures. In the barn's frame the fast moving pole is contracted along its direction of travel, so for a moment both ends sit inside the barn and both doors could shut at once. In the pole's own frame it is the barn that rushes past and contracts, so the pole never fits. Neither observer is wrong, because the two frames disagree about whether the two doors close at the same time, which is the relativity of simultaneity. The calculator reports the contracted pole length seen from the barn, the Lorentz factor gamma, the contracted barn length seen from the pole, and whether the pole fits in the barn frame. It is a teaching tool for physics students meeting length contraction and simultaneity for the first time, using speeds close to light where the effects become large.
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m
m
v / c
8.72 m
pole length measured in the barn's frame
Lorentz factor γ2.294
Barn in pole's frame4.36 m
Fits in barn frameYes
In the barn frame the pole contracts to 8.72 m and fits inside the 10 m barn; in the pole's frame the barn contracts to 4.36 m and the 20 m pole cannot fit. Both frames are correct.
How it works
The Lorentz factor gamma is 1 divided by the square root of 1 minus the speed squared, where the speed is written as a fraction of the speed of light. A length measured in a frame where it is moving is its rest length divided by gamma, so the faster the motion the shorter it appears along the direction of travel. The barn measures the pole contracted; the pole measures the barn contracted. The pole fits in the barn frame when the contracted pole length is less than or equal to the barn's rest length. The speed must be below the speed of light.
Worked example
A 20 m pole runs through a 10 m barn at 0.9 c. Gamma is 1 divided by the square root of 1 minus 0.81, which is 1 divided by the square root of 0.19, about 2.294. The pole seen from the barn is 20 divided by 2.294, which is 8.72 m, and that is less than 10 m, so it fits in the barn frame. Seen from the pole, the barn is 10 divided by 2.294, which is 4.36 m, and the 20 m pole clearly does not fit. The disagreement is real and comes from simultaneity.
