Last updated:

# Length Contraction Calculator

Table of contents

Length is relative: ladder paradoxLength contraction formula

You might have heard about the concept of length relativity or length contraction. Whether you are already familiar with the phenomenon of length relativity or just taking your first steps in special relativity, this length contraction calculator will help you better understand relativistic effects and the length contraction equation. The length contraction is also known as Lorentz contraction.

💡 We also have other relativity calculators, such as the relativistic kinetic energy calculator and the gravitational time dilation calculator.

## Length is relative: ladder paradox

The length contraction is a relativistic phenomenon that a length of a moving object measures shorter than its length when measured at rest. This effect is only noticeable at enormous velocities.

The length contraction leads to a ladder paradox. The paradox involves a garage and a ladder that should be placed inside the garage. The ladder, however, is too long to fit in.

The decision is to run with the ladder. From the point of view of a stationary observer, the ladder will then appear shorter and could fit in the garage.

However, the garage is moving for a person running with the ladder. Therefore its length is contracted, and it cannot accommodate the ladder. Who is right?

The resolution of the paradox comes from noticing that the ladder fits in the garage if the following two events happen at the same time:

1. The front of the ladder is in the garage.
2. The back of the ladder is in the garage.

But whether two events occur at the same time depends on the point of view. These two events might happen simultaneously for a person observing the scenario, and this person can claim that the ladder fits in. For a person running with the ladder, these two events do not happen at the same time, and the ladder does not fit in.

## Length contraction formula

How fast do you need to run to confuse your friends? You can calculate the length contraction with the following equation:

$\small L = \frac{1}{\gamma(v)}L_0 = L_0 \sqrt{1 - v^2/c^2},$

where:

• $L$ – Observed length of the object in motion;
• $L_0$ – Length of the object in its rest frame, also known as proper length;
• $v$ – Speed of the object;
• $c$ – Speed of light (299,792,458 m/s); and
• $\gamma(v)$ – Lorentz factor, defined as $\gamma(v) \equiv 1/\sqrt{1 - v^2/c^2}$.

Or you can use this length contraction calculator.

For the effect of length contraction to be noticeable, the speed must be very high – comparable to the speed of light. We do not have experience in our daily life with objects moving at such high velocities. That makes the effects of special relativity very counterintuitive.

The effects of are challenging to observe because, in addition to the high speed required, it is difficult to measure a length of an object moving with respect to us. However, there are indirect confirmations of the length contraction. For example, the Earth is bombarded with muons created in the upper atmosphere from collisions of cosmic rays with atoms.

Muons have an extremely short lifetime; they decay quickly into other elementary particles before reaching the Earth's surface. However, as they move with a velocity close to the speed of light, the distance from the upper atmosphere to the Earth's surface is (for them) contracted, and we can observe muons at Earth's surface.

If you want to learn about or calculate other effects of special relativity, try our E = mc2 calculator or time dilation calculator.