Paper 4, Section II, B

Dynamics and Relativity | Part IA, 2013

A frame SS^{\prime} moves with constant velocity vv along the xx axis of an inertial frame SS of Minkowski space. A particle PP moves with constant velocity uu^{\prime} along the xx^{\prime} axis of SS^{\prime}. Find the velocity uu of PP in SS.

The rapidity φ\varphi of any velocity ww is defined by tanhφ=w/c\tanh \varphi=w / c. Find a relation between the rapidities of u,uu, u^{\prime} and vv.

Suppose now that PP is initially at rest in SS and is subsequently given nn successive velocity increments of c/2c / 2 (each delivered in the instantaneous rest frame of the particle). Show that the resulting velocity of PP in SS is

c(e2nα1e2nα+1)c\left(\frac{e^{2 n \alpha}-1}{e^{2 n \alpha}+1}\right)

where tanhα=1/2\tanh \alpha=1 / 2.

[You may use without proof the addition formulae sinh(a+b)=sinhacoshb+coshasinhb\sinh (a+b)=\sinh a \cosh b+\cosh a \sinh b and cosh(a+b)=coshacoshb+sinhasinhb\cosh (a+b)=\cosh a \cosh b+\sinh a \sinh b.]

Typos? Please submit corrections to this page on GitHub.