Diagram

\[\tau = h/c\]\[\beta = V/c\]

\[s=\sqrt{h^2+V^2t^2}\]\[\Delta t=s/c=\sqrt{\tau^2+\beta^2t^2}\]

\[t'=t+\Delta t=t+\sqrt{\tau^2+\beta^2t^2}\]

\[t'^2-2t't+(1-\beta^2)t^2-\tau^2=0\]

\[\choise{\beta=1\ \&\ t'\gt0&t=\frac{t'^2-\tau^2}{2t'}\\ \beta\lt1&t=\frac{1}{1-\beta^2}\left(t'-\sqrt{\beta^2t'^2+(1-\beta^2)\tau^2}\right)\\ \beta\gt1\ \&\ t'\gt\sqrt{1-1/\beta^2}\tau&t=\frac{1}{\beta^2-1}\left(-t'\pm\sqrt{\beta^2t'^2-(\beta^2-1)\tau^2}\right)}\]

\[P=\frac{\dd{E}}{\dd{t}},\ P'=\frac{\dd{E}}{\dd{t'}},\ \frac{P'}{P}=\frac{\dd{t}}{\dd{t'}}\]

\[\frac{\dd{t'}}{\dd{t}}=1+\frac{\beta^2t}{\sqrt{\tau^2+\beta^2t^2}}\]

\[t\lt-\frac{\tau}{\beta\sqrt{\beta^2-1}}\]