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Say we have two points, A and B, along the x-axis, and a spacecraft is moving past the points in the positive direction of the x-axis.
The proper length of the line segment AB is L0. It is a proper length since the points A and B are fixed in the frame of reference of the x-axis. For our muon, A could be the point where it was created, and B might be the ground. The velocity of the spacecraft (or muon) is now
Here, t is the time according to the frame of reference of the points A and B. From the standpoint of the spacecraft, points A and B are rushing towards it with the velocity v, so we get
The travel along the length L, between A and B, will, according to the spacecraft, take the proper time t0. But we know from the time dilation that t=λt0, and thus that
This is the formula for the length contraction.
- Continuing with the moun problem from the previous page. Say the ground is 10 km down.
- How far down would that be according to the frame of reference of the muon travelling at 0.9992c?
- How long time would it take for the muon to reach the ground, as seen from the frame of reference of the muon?
- Will it reach the ground?
- What speed would a spaceship have to have to be seen as half as long as its proper length?

Previous page : Time dilation
