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Length Contraction Explained: Formula & Example

Length Contraction Explained: Formula & Example
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  • 0:00 Light Speed Physics
  • 1:44 Preparing for Length…
  • 4:35 Length Contraction
  • 6:11 Shrinking Ship Example
  • 7:29 Lesson Summary
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Lesson Transcript
Instructor: Ryan Hultzman
Learn what length contraction has to do with the speed of light, and why we don't notice it in every day life. Also learn a little about other relativity topics that will help you understand length contraction better.

Light Speed Physics

Not all physics phenomena are readily apparent to the naked eye. Some of these phenomena only become apparent in extreme conditions. Others are always affecting the world around us but cannot normally be observed by us. Length contraction is one that's always working but we don't necessarily notice it.

When an object is moving, its length shrinks in the same direction as its velocity. This is called length contraction. This affects all objects at all times when they are moving. It affects a race car going around a track, you when you're running, and even a snail slowly inching its way along the ground. You might be thinking this sounds ridiculous since you've never noticed anything shrinking as it moved. That's because under normal circumstances the effect is so incredibly tiny that it's not noticeable at all. An object needs to be moving at speeds approaching the speed of light before length contraction starts to take a great affect on it.

To put things in perspective, let's look at one of the fastest moving vehicles ever created - the space shuttle. When in orbit a space shuttle can move at speeds of around 17,500 mph. The speed of light is 3 x 10^8 meters per second, or 6.7 x 10^8 mph. The space shuttle is moving at roughly 0.0026% the speed of light. That is too slow for length contraction to have a noticeable effect on the shuttle, and it's one of the fastest moving objects ever created. Even though length contraction is always occurring, it seems so hard to believe because we never come across objects in our daily lives moving fast enough for its effects to be apparent.

Preparing for Length Contraction

To dive deeper into how length contraction works, there are three very important concepts you need to know. First is a reference frame, or frame of reference, which can be thought of as an observer's point of view. Imagine a person sitting in a train as it moves past another person standing still next to the tracks. For the person standing still outside the train, their frame of reference has the train moving past them at some velocity. For the person in the train, from their frame of reference the train is standing still, and the person outside the window is moving past.

The next concept is the second postulate of special relativity. It states that the speed of light in a vacuum is constant for all inertial reference frames. An inertial reference frame is a reference frame in which an object remains at rest, or moves in a straight line unless acted upon by another force. No matter from what point of view someone views light in a vacuum, it will always be moving at 3 x 10^8 m/s in any inertial reference frame.

Finally there is time dilation. Time dilation states that time for an object moving runs at a slower rate than an object standing still. This phenomenon goes hand-in-hand with length contraction since both are always affecting any moving object, and both are only noticeable at speeds approaching the speed of light.

Like the train earlier, imagine a person on a spaceship moving at a constant velocity close to the speed of light, and a person outside the ship standing still. In our ship, there are two mirrors that bounce a light ray back and forth in a vacuum. From the point of view of the observer inside the ship, the light ray bounces straight up and down as seen in this image:

Time Dilation Diagram

From the point of the observer outside the ship the light ray is acting much differently. Since the ship is moving the light ray must also be moving in the same direction while bouncing back and forth as seen in the image here:

Time Dilation Diagram 2

To the observer outside the ship, the light ray is moving over a much larger distance than the observer inside sees. However, according to the second postulate of relativity the light ray from both frames of reference is traveling at the same speed. Since the velocity of light is the same from both frames of reference, but the distance the ray of light travels is different, it turns out that the rate of time passing must be different for each observer. The observer on the outside will see time passing more slowly for the observer inside the ship. The mathematical relationship for time dilation is given by the following formula:

t = t{0} * gamma

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