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###### Length contraction

Physics (Year 12) - Special Relativity

Dev Lohar

# Equation

The equation for length contraction is as follows:

Sometimes the equation is simplified to:

As the name suggests, the observed length, l, appears to be less (ie. contracted) than the proper length. Itâ€™s important to not get it confused with time dilation in which the observed time, t, appears to be higher (ie. dilated) than the proper time.

# Understanding

The length contraction effect is only applicable for lengths parallel to the direction of travel. Imagine youâ€™re standing on a platform at a train station and youâ€™re facing the train tracks so that the train track is perpendicular to you. When a train travels along the track, you will observe that the train seems to be shorter lengthwise than it was when at rest. Other dimensions of the train such as its width or height do not change and appear to be the same as when the train is at rest. Therefore, the length in the direction of motion is contracted.

# Example

In the section about time dilation, there was an example on how the muon lifetime is dilated from the frame of reference of an observer on Earth, and hence muons are able to be detected on Earth. However, from the frame of reference of muons, they experience length contraction. From the muonsâ€™ frame of reference, they are stationary and space around them is moving past them at a given speed v. Hence, the distance from the upper atmosphere (where muons are created) to the Earth surface is contracted. Therefore, in the frame of reference of the muons they have travel a lower distance and so are able to reach the Earthâ€™s surface. To understand this further, imagine the distance between the upper atmosphere and the Earthâ€™s surface is a ladder which the muons climb down on. From the frame of reference of an observer on Earth, they would observe the muons climbing down at a speed v but from the frame of reference of the muons, the ladder is moving past them at speed v. Since we know that only the length of an object in the direction of motion is contracted, the length of the ladder will be contracted and therefore the muons are able to reach the Earthâ€™s surface because they have â€˜lessâ€™ distance to travel.

Another common example is the train in a tunnel example. Imagine a train which is longer than a tunnel as seen in the diagram below.

*diagram*

When the train is stationary, it is observed that the train does not fit inside the tunnel. But when the train is moving at a sufficient speed, enough for relativistic effects to take place, it is observed from the frame of reference of a stationary observer that the train does fit in the tunnel because the length of the train is contracted.

*diagram*

Hence the train both does and doesnâ€™t fit in the tunnel.

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