Length Contraction
The length of any object in a moving frame will appear foreshortened in the direction of motion, or contracted. The amount of contraction can be calculated from the Lorentz transformation.
Special relativity states that the distance between two points can differ in different reference frames. The distance between points, and therefore the length, depends on the velocity of one reference frame relative to another. In one reference frame, an object being measured will be at rest. This is called the proper length, and is labeled Δl0. In another reference frame, an observer will see the object moving. The length of the object in this reference frame is observed length, and is labeled Δl. The observed length is always shorter than the proper length. This effect is called length contraction. Both Δl0 and Δl are measured in meters (m).
\[Observed \: length = (proper \: length) \sqrt{ 1- \left (\frac{velocity}{speed \: of \: light} \right )^{2}} \]
\[\Delta l=\Delta l_{0}\sqrt{1-\left ( \frac{v}{c} \right )^{2}} \]
Δl = the observed length, in the reference frame in which the object is moving (m)
Δl0 = the proper length, in the reference frame in which the object is at rest (m)
v = velocity (m/s)
c = speed of light (3.0 x 108 m/s)
let value of Δl0 = a, Δl=b
