# A Roller Coaster Ride through Relativity/Appendix B

## Length Contraction

Using the context of the river race, if we see Albert and Beatrice arrive back at exactly the same instant and knowing that Beatrice cannot row faster than Albert, we must conclude that Beatrice has a shorter distance to row.

How much shorter? Suppose that Albert rows a distance lA and Beatrice rows lB.

${\displaystyle {\text{Time for Albert to finish}}={2l_{A} \over {\sqrt {c^{2}-v^{2}}}}}$
${\displaystyle {\text{Time for Beatrice to finish}}={l_{B} \over c-v}+{l_{B} \over c+v}={2l_{B}c \over c^{2}-v^{2}}}$

Now these must be equal so:

${\displaystyle {2l_{A} \over {\sqrt {c^{2}-v^{2}}}}={2l_{B}c \over c^{2}-v^{2}}}$

from which we get:

${\displaystyle l_{B}={\sqrt {1-v^{2}/c^{2}}}\,l_{A}}$

What this means is that from the point of view of a stationary observer, a rod of proper length l0 (ie whose length is l0 when at rest) will have a length l given by

${\displaystyle l={\sqrt {1-v^{2}/c^{2}}}\,l_{0}=l_{0}/\gamma }$

when it moves with a velocity v in a direction parallel to its length. (It will always appear to have the same length l0 perpendicular to the direction of motion.)

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