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messier992

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## Homework Statement

Three trains A, B, and C with equal proper lengths L are moving on parallel tracks. In the frame of A, B has length 4L/5. In the frame of A, what is the speed of C, if the lengths of A and B are equal in the frame of C

## Homework Equations

u'=[u-v]/[(1-uv)]

c=1

## The Attempt at a Solution

L' = L/γ => 4L/5 = L*sqrt(1-u^2) => sqrt(1-(4/5)^2) = u => sqrt[(25-16)/25] = u => u= 3/5

Where:

u'=v since is the relative speed of C/B or C/A since B/C=A/C. This is true because L is the same for all trains, and both trains A and B have the same length in C?

v = relative speed of C/A

u = relative speed of B/A

Thus

v=[u-v]/[(1-uv)] => v*(1-uv)=[u-v] => v-v^2*u-u+v=0 => -v^2+(2v/u)-1 = 0 => v^2-(10v/3) +1 = 0

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