Ladder and Barn Paradox Calculator

A ladder too long for a barn fits inside it when it runs fast enough — and does not, in its own frame, at the same time. Both frames are right. This page takes your ladder, your barn and your speed and tells you what each frame sees, and why the disagreement is about simultaneity rather than about length.

Last updated 11 Nov 2026

A ladder, a barn, a speed, and whose story you want

Resolving it is the page, not a number

Rest length of the ladderIn metres, as the ladder measures itself. For the classic version of the puzzle make this LONGER than the barn, which is what creates the apparent contradiction. The default is a 10 m ladder.

Rest length of the barnIn metres, as the barn measures itself. The barn is the thing with two doors, one at each end, which can be shut and reopened. The default is 8 m, so the ladder does not fit while either of them is standing still.

Speed of the ladder, as a fraction of cBetween 0 and 1. The ladder’s contracted length in the barn’s frame is lad × √(1 − β²), so for a 10 m ladder and an 8 m barn you need β above 0.6 before it fits. Typing 1 or more is refused rather than answered, because nothing with mass can be given that speed.

Whose story do you wantThe third option is the one worth reading. The other two give you a single frame’s honest account, which is useful when a question names a frame and wants that frame’s answer only.

The same ladder and barn drawn twice, each in its own frame, both to the same scale in metres. In the upper panel the barn is at rest and the ladder is contracted; the two red dots sit at the SAME height, because in that frame the doors shut at the same instant. In the lower panel the ladder is at rest and the barn is contracted; the two red dots are at DIFFERENT heights, because in that frame the far door shuts first and the near door later. The dashed blue box marks the frame you asked for. Nothing about the dots is decoration: the difference between them IS the resolution.

Barn frame: it fits. Ladder frame: it never does. Both are correctExample

A 10 m ladder through an 8 m barn at 0.8c, read in both frames

Two contractions and one disagreement about ‘at the same time’

barn’s frame: L_ladder = L₀√(1 − β²) · ladder’s frame: L_barn = B₀√(1 − β²) · Δt′ = −γvΔx/c²

L₀√(1−β²): the ladder as the BARN measures it. Fits when this is at most the barn’s rest length. For a 10 m ladder and an 8 m barn that needs β of at least 0.6
B₀√(1−β²): the barn as the LADDER measures it. Both objects are contracted, each in the other’s frame, and there is no contradiction in that because they are two different measurements made by two different observers
Δt′ = −γvΔx/c²: THE KEY LINE. Two events separated by Δx and simultaneous in one frame are separated in time by this much in a frame moving at v. For the two doors Δx is the barn’s length, so the gap between the closings in the ladder’s frame is γvB₀/c², and it is not zero
what both frames agree on: whether the ladder was hit by a door, whether it left the barn intact, and the order of events at ONE place. Frames never disagree about those. They disagree only about the ordering of events far apart

Worked example

A 10 m ladder through an 8 m barn at 0.8c, read in both frames
THE BARN'S FRAME. γ = 1/√(1 − 0.64) = 1/0.6 = 1.667, so the ladder is contracted to 10 × 0.6 = 6 m. The barn is 8 m. The ladder fits with 2 m to spare, and both doors can be shut at the same instant with it inside.
THE LADDER'S FRAME. Nothing has happened to the ladder — it is 10 m, as it always was. The barn is now the thing moving, so the BARN is contracted to 8 × 0.6 = 4.8 m. A 10 m ladder cannot be inside a 4.8 m barn at any moment.
SO WHO IS RIGHT? Both. The question “is the whole ladder inside the barn?” secretly means “are both ends of the ladder between the doors AT THE SAME TIME?”, and the words “at the same time” are frame-dependent.
WORK OUT THE TIME GAP. The two door-closings are Δx = 8 m apart and simultaneous in the barn's frame. In the ladder's frame they are separated by γvΔx/c² = 1.667 × 0.8 × 8 / 299 792 458 = 3.56×10⁻⁸ s, about 36 nanoseconds. The far door shuts and reopens 36 ns before the near door shuts.
CHECK THAT NOTHING PHYSICAL IS IN DISPUTE. In both accounts the ladder emerges undamaged, no door ever strikes it, and each individual door is observed to close and open in the same order by everybody. Those are the frame-independent facts and they match. What differs is only the ordering of two events in different places.
THE GENERAL LESSON. Whenever a relativity problem seems to produce a contradiction, look for the place where you assumed two distant events happened at the same time. That assumption is almost always the thing that is frame-dependent, and almost always the thing that dissolves the paradox.

What each frame says, at a few speeds

Read the third and fourth columns together. **Both objects are contracted, each in the other’s frame**, and that is not a contradiction but the whole point: contraction is something one observer measures about another’s object, not something that happens to the object. The disagreement about fitting only begins in the fourth row, where the contracted ladder has become short enough for the barn while the ladder’s own view is unchanged.
β	γ	Ladder, in the barn’s frame	Barn, in the ladder’s frame	Verdict
0	1	10 m	8 m	does not fit; both frames agree
0.4	1.091	9.17 m	7.33 m	does not fit; both frames still agree
0.6	1.25	8.00 m	6.40 m	exactly fits in the barn’s frame — the threshold
0.8	1.667	6.00 m	4.80 m	fits in the barn’s frame, never in the ladder’s
0.9	2.294	4.36 m	3.49 m	the same disagreement, more comfortably
0.99	7.089	1.41 m	1.13 m	the ladder is now a seventh of its own length

They disagree about simultaneity, not about the ladder

The setup. A 10 m ladder runs at 0.8c towards an 8 m barn with a door at each end. In the barn’s frame the ladder is contracted to 6 m, so there is a moment when it is entirely inside and the farmer can shut both doors at once. In the ladder’s frame the ladder is still 10 m and the barn has contracted to 4.8 m, so the ladder is never inside. Both statements are made correctly by competent observers using correct physics, and they cannot both be describing the same fact. So they are not: one of them is not a fact about the world at all.

The resolution is one sentence. “The whole ladder is inside the barn” means “both ends of the ladder are between the doors at the same time”, and simultaneity is frame-dependent. Two events happening in different places can be simultaneous for one observer and not for another. The two door-closings are 8 m apart; in the barn’s frame they happen together, and in the ladder’s frame they are 36 nanoseconds apart. The far door shuts and reopens while the front of the ladder has already gone past it, and the near door shuts later, once the back has gone in. Each door closes with nothing in the way. Nobody is lying and nothing is broken.

What the two frames do agree about. This is the part worth being able to state. Any question whose answer is a single physical fact gets the same answer in every frame: was the ladder damaged, did a door strike it, did it come out of the far end, in what order did a particular door close and open. Frames never disagree about events at the same place, or about causally connected events. What they can disagree about is the ordering of two events far apart with no causal link between them — and the doors, 8 m and 36 ns apart, are exactly that.

Where this goes wrong. Saying one frame is “really” right — usually the barn’s, because it feels stationary. It is not; there is no such thing as really stationary. Saying the contraction is an optical illusion: it is a measurement, and it survives after the light-travel delays are taken out. Contracting both objects in the same frame, which is the commonest algebraic error here — in the barn’s frame the ladder is short and the barn is its full length, and in the ladder’s frame the opposite. And trying to resolve the puzzle by arguing about whether the ladder is “really” long or short, when the disputed word in the question was never a length word; it was “simultaneously”.

Frequently asked questions

Does the ladder actually fit, yes or no?

The question has no frame-free answer, which is the finding rather than a dodge. In the barn’s frame there is an instant at which the whole ladder lies between the two closed doors, and that is a real measurement, not an appearance. In the ladder’s frame there is no such instant. “At one instant” is the part that is frame-dependent, so the question as asked is like asking whether two events in different cities happened at the same time without saying by whose clock.

What if the doors stay shut and the ladder is trapped?

Then the frames still agree about the physical outcome, because the far door now hits the ladder. In the barn’s frame the ladder is inside when the doors shut and then crashes into the closed far door. In the ladder’s frame the far door shuts while the ladder’s front is already through it, so the door strikes the ladder immediately — and the shock cannot travel to the back of the ladder faster than light, so the back is still moving while the front has stopped. The ladder is compressed. Both frames agree it is compressed and that it ends up inside; they describe the sequence differently.

Why can’t I just say the barn’s frame is the real one?

Because there is no experiment that would identify it as real. The first postulate of relativity is that the laws of physics are the same in all inertial frames, so no frame can be picked out as the true one. The barn feels stationary to you only because you are standing in it; from the ladder’s point of view the barn is the thing hurtling past at 0.8c, and that description is exactly as valid.

How big is the time gap between the two door closings?

In the ladder’s frame it is γvΔx/c², where Δx is the barn’s rest length. For an 8 m barn at 0.8c that is about 36 nanoseconds. It is small, but it is not an approximation or a rounding artefact — it is exactly the amount of non-simultaneity needed to make the two accounts consistent, and if you work the problem through with the Lorentz transformation you get the same number from the algebra.

Is this the same as the twin paradox?

Different puzzle, same underlying machinery. The twin paradox is about which of two clocks reads less when they are brought back together, and is resolved by noticing that only one twin accelerated; see the space travel page for the numbers. The ladder and barn is about whether two distant events happen at once, and is resolved by the relativity of simultaneity. Both are cases of the same lesson: a quantity you assumed was absolute turns out to depend on the frame.

Will this be on my exam?

Possibly as a discussion question rather than a calculation. AQA’s special relativity section asks for time dilation, length contraction and mass-energy, not for this paradox by name — but a question on length contraction that asks you to explain how two observers can disagree is this problem in disguise. The CBSE Class XII curriculum has no relativity at all, so on that syllabus it is reading for interest.

Related calculators

References

A. Einstein, “Zur Elektrodynamik bewegter Körper”, Annalen der Physik 17 (1905) 891–921 — the paper in which the two postulates, the Lorentz transformation, time dilation and length contraction all appear together, and the companion three-page note “Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?”, Annalen der Physik 18 (1905) 639–641, which adds the mass-energy result. Cited for the result, not for a number: neither paper was read for this build, and nothing on this page rests on a figure taken from them.
AQA, AS and A-level Physics (7407, 7408) specification, section 3.12.3 Special relativity, inside the option 3.12 Turning points in physics (A-level only). Read from aqa.org.uk on 11 November 2026. It is worth knowing exactly what a board asks for: 3.12.3.3 requires proper time and time dilation in the form t = t₀ / √(1 − v²/c²), with evidence from muon decay; 3.12.3.4 requires length contraction as l = l₀√(1 − v²/c²); and 3.12.3.5 requires mass-energy equivalence, the form E = m₀c²/√(1 − v²/c²), graphs of mass and kinetic energy against speed, and Bertozzi’s experiment as direct evidence. Note what that implies: AQA still teaches relativistic mass. This page cites the specification; it does not claim the board endorses it.
CODATA Internationally Recommended 2022 Values of the Fundamental Physical Constants, NIST Standard Reference Database 121. Read on 11 November 2026 from the machine-readable table at physics.nist.gov/cuu/Constants/Table/allascii.txt, which prints “speed of light in vacuum 299 792 458 (exact)” — exact because the 1983 definition of the metre fixes it, so c² = 8.9875517873681764×10¹⁶ m²/s² carries no uncertainty either. The same table gives G = 6.674 30(15)×10⁻¹¹ m³kg⁻¹s⁻², which is the least precisely known of the constants on this page by a factor of ten thousand, standard acceleration of gravity 9.806 65 (exact), and the rest energies used here: electron 0.510 998 950 69(16) MeV, proton 938.272 089 43(29) MeV, muon 105.658 3755(23) MeV.
CBSE / NCERT, Physics, Class XII, Curriculum 2025–26, read on 11 November 2026 from the government-hosted syllabus at edustud.nic.in (SYLLABUS_2025_26/12/12_Physics_EM.pdf). The nine units are Electrostatics, Current Electricity, Magnetic Effects of Current and Magnetism, Electromagnetic Induction and Alternating Currents, Electromagnetic Waves, Optics, Dual Nature of Radiation and Matter, Atoms and Nuclei, and Electronic Devices. Special relativity, time dilation, length contraction and the Lorentz factor appear nowhere in it. The only trace of the subject is “mass-energy relation, mass defect” in Chapter 13, Nuclei — and the document does not even write E = mc² out. So if you are on this syllabus, this topic is reading beyond it rather than revision for it, which is a useful thing to know the night before a paper.
Particle Data Group, Review of Particle Physics — Summary Table of Leptons, read on 11 November 2026 from pdg.lbl.gov/2025/tables/rpp2025-sum-leptons.pdf. Muon mean life τ = (2.196 981 1 ± 0.000 002 2)×10⁻⁶ s; muon mass 105.658 3755 ± 0.000 0023 MeV. It is a MEAN life, not a half-life: the half-life is τ ln 2 = 1.5230 µs, and quoting one where the other is meant is the commonest error in a muon question. cτ = 658.64 m, which is the distance a muon would cover in one mean lifetime if there were no dilation at all.