A rod flies with constant velocity past a mark which is stationary in the reference frame K. In the frame K it takes `Deltat=20ns` for the rod to fly past the mark. In the reference frame fixed to the rod the mark moves past the rod of `Deltat^'=25ns`. Find the proper length of rod.

Explanation

In the frame K the length l of the rod is related to the time of flight `Deltat` by
`l=vDeltat^'`
But `l_0=(l)/(sqrt(1-beta^2))=(vDeltat)/(sqrt(1-beta^2))`, `beta=v/c`
Thus, `vDeltat^'=(vDeltat)/(sqrt(1-beta^2))`
So `1-beta^2=((Deltat)/(Deltat^'))^2` or `v=csqrt(1-((Deltat)/(Deltat^'))^2)`
and `l_0=csqrt((Deltat^')^2-(Deltat)^2)=cDeltat^'sqrt(1-((Deltat)/(Deltat^'))^2)`