The reference frame `K^'` moves in the positive direction of the x axis of the frame K with a relative velocity `V^'`. Suppose that at the moment when the origins of coordinates O and `O^'` coincide, the clock readings at these points are equal to zero in both frames. Find the displacement velocity x of the point (in the frame K) at which the readings of the clocks of both reference frames will be permanently identical. Demonstrate that `overset(.)x lt V`.

Explanation

Suppose `x(t)` is the locus of points in the frame K at which the readings of the clocks of both reference system are permanently identical, then by Lorentz transformation
`t^'=(1)/(sqrt(1-V^2//c^2))(t-(Vx(t))/(c^2))=t`
So differentiating `x(t)=c^2/V(1-sqrt(1-V^2/c^2))=c/beta(1-sqrt(1-beta^2)), beta=V/c`
Let `beta=tan h theta, 0 le theta lt oo`, Then
`x(t)=(c)/(tanhtheta)(1-sqrt(1-tan h^2theta))=c(coshtheta)/(sinhtheta)(1-(1)/(coshtheta))`
`=c(coshtheta-1)/(sinhtheta)=csqrt((coshtheta-1)/(coshtheta+1))=ctanhtheta/2lev`
(`tan htheta` is a monotonically increasing function of `theta`)