A rod moves along a ruler with a constant velocity. When the positions of both ends of the rod are marked simultaneously in the reference frame fixed to the rule, the difference of readings on the rule is equal to `Deltax_1=4.0m`. But when the positions of the rod's ends are marked simultaneously in the reference frame fixed to the rod, the difference of readings on the same ruler is equal to `Deltax_2=9.0m`. Find the proper length of the rod and its velocity relative to the ruler.

Explanation

In the reference frame fixed to the ruler the rod is moving with a velocity v and suffers Lorentz contraction. If `l_0` is the proper length of the rod, its measured length will be
`Deltax_1=l_0sqrt(1-beta^2), beta=v/c`
In the reference frame fixed to the rod the ruler suffers Lorentz contraction and we must have
`Deltax_2sqrt(1-beta^2)=l_0` thus `l_0=sqrt(Deltax_1Deltax_2)`
and `1-beta^2=(Deltax_1)/(Deltax_2)` or `v=csqrt(1-(Deltax_1)/(Deltax_2))`