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# Special Theory of Relativity

## Interpret motion between different objects that are moving at constant speeds relative to each other.

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Length Contraction

### Length Contraction

Credit: Gary & Anna Sattler
Source: http://www.flickr.com/photos/9512074@N02/815794802
License: CC BY-NC 3.0

Can you fit a car that is 6.0 ft long into a garage that is 5.5 ft long if it moves at relativistic speeds? Read on to find out.

#### Amazing But True

• In everyday life, when an object travels at a given speed, it is expected that its dimension are independent of whether the object is moving or not. Even though we can’t see a car being compressed as it drives by us, it actually does a very, very small amount. When any object travels at a non-zero velocity, length contraction is seen when the length of an object is measured by an outside observer. Even though an object’s length is being contracted, it is very difficult to see unless the object has a velocity that is a large fraction of the speed of light.
• Also known as Lorentz contraction, the length an object will be observed to be contracted by is given as

\begin{align*}L=L_o\sqrt{1 - \frac{v^2}{c^2}}\end{align*}

where \begin{align*}L\end{align*} is the length observed by an observer relative to the motion of the object, \begin{align*}L_o\end{align*} is the proper length, or the length of the object in its rest frame and \begin{align*}v\end{align*} is the velocity of the object between the observer and the moving object.

• The above equation shows that as an object is traveling along one axis at a fraction of the speed of light, its length along that axis will be contracted by a given amount.

#### Show What You Know

Using the information provided above, answer the following questions.

1. What speed would be required for the car in the picture to be at the correct length to fit inside of the garage?
2. Does the car actually fit inside the garage?
3. How could the vertical height of the car be contracted as well?

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### Image Attributions

1. [1]^ Credit: Gary & Anna Sattler; Source: http://www.flickr.com/photos/9512074@N02/815794802; License: CC BY-NC 3.0

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