A particle of mass $M$ is at rest at $x=0$, in coordinates $(t, x)$. At time $t=0$ it decays into two particles $\mathrm{A}$ and $\mathrm{B}$ of equal mass $m<M / 2$. Assume that particle A moves in the negative $x$ direction.

(a) Using relativistic energy and momentum conservation compute the energy, momentum and velocity of both particles $A$ and $B$

(b) After a proper time $\tau$, measured in its own rest frame, particle A decays. Show that the spacetime coordinates of this event are

$$\begin{aligned} t &=\frac{M}{2 m} \tau \\ x &=-\frac{M V}{2 m} \tau, \end{aligned}$$

where $V=c \sqrt{1-4(m / M)^{2}}$.