Barn-Pole Paradox

Enter the pole's rest length, the barn's rest length, and the pole's speed as a fraction of light speed to see the Lorentz-contracted length and whether the pole fits inside the barn.

Quick Facts

Formula
L' = L₀ / γ, where γ = 1 / √(1 − v²/c²)
Length contraction only applies along the direction of motion, and only in the frame doing the measuring.

Your Results

Calculated
Lorentz factor (γ)
-
Length/time scaling at this speed
Contracted pole length
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Pole length as measured in the barn's frame
Clearance in barn frame
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Barn length minus contracted pole length
Critical speed to fit
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Minimum v/c for the pole to just fit

Ready

Enter both rest lengths and a speed, then press Calculate.

About the Barn-Pole Paradox

The barn-pole (or "ladder") paradox is a classic thought experiment in special relativity. A pole longer than a barn is carried through the barn at relativistic speed. According to an observer standing next to the barn, length contraction shrinks the moving pole enough that it briefly fits inside with both doors shut. According to an observer riding along with the pole, it is the barn that contracts, so the pole never fits and always sticks out of one door. Both descriptions are correct in their own reference frame — the apparent contradiction is resolved by the relativity of simultaneity, since "both doors closed at once" is not the same event for both observers.

The formula

This calculator uses the standard special-relativity length contraction equation:

L' = L₀ / γ, where the Lorentz factor is γ = 1 / √(1 − v²/c²)

Here L₀ is an object's proper (rest-frame) length, v is the relative speed, c is the speed of light, and L' is the length measured by an observer for whom the object is moving. Applied to this paradox: the pole's rest length L₀ contracts to L₀/γ as measured by the barn observer. If that contracted length is no longer than the barn's rest length L_b, the pole fits — in the barn's frame. The calculator also reports the critical speed fraction, found by solving L₀/γ = L_b for v/c: β_critical = √(1 − (L_b/L₀)²). Any speed at or above this value makes the pole fit as measured from the barn.

Working with the inputs

  • Enter both lengths in the same unit (meters here); the ratio L_b/L₀ is all that matters for the fit calculation, so consistent units are essential.
  • Speed is entered as a fraction of light speed, v/c, also called β. It must satisfy 0 ≤ β < 1 — nothing with a rest length can reach or exceed c.
  • If the pole is already shorter than or equal to the barn at rest (L₀ ≤ L_b), it fits at any speed, including zero, and the critical speed is reported as 0.

Knowing the limits

This is a kinematic, frame-dependent statement about measured length, not a claim about physical compression. Length contraction is only along the direction of motion, applies symmetrically (each frame sees the other's ruler as short), and the "doors closing simultaneously" only holds in one frame at a time. The calculator reports the fit as seen in the barn's rest frame; the pole's own frame would report the opposite fit, which is the heart of the paradox rather than an error.

Frequently Asked Questions

What is the Barn-Pole (Ladder) Paradox?
It is a thought experiment in special relativity: a pole longer than a barn is run through it at relativistic speed. In the barn's rest frame, length contraction shrinks the moving pole enough that it briefly fits inside with both doors shut. In the pole's own rest frame, the barn is the one that contracts, so the pole never fits. Both views are correct; the paradox is resolved by the relativity of simultaneity — the two doors do not close at the same instant in every frame.
What formula does this calculator use?
It uses the special-relativity length contraction formula L' = L₀ / γ, where γ = 1 / √(1 − v²/c²) is the Lorentz factor. The pole's rest length is contracted by γ as measured in the barn's frame, then compared to the barn's rest length to determine whether it fits.
How is the critical (minimum) speed calculated?
The critical speed is the slowest speed at which the contracted pole length exactly equals the barn length: L₀/γ = L_b. Solving for β = v/c gives β_critical = √(1 − (L_b/L₀)²). Any speed at or above this value makes the pole fit in the barn's frame; if the pole is already shorter than the barn at rest, no motion is needed.
Does the pole really fit, or is that just an illusion?
Both frames are physically valid, and there is no true contradiction. Length contraction and simultaneity are frame-dependent, so "fits" only means fits according to a specific observer's set of simultaneous events. Neither the pole nor the barn is physically crushed or altered; only the measured length in each frame differs.

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