= Solution
Before its second decay A moves uniformly with $x=-Vt$. Its <Lorentz factor> is
$$
\gamma_A=\frac1{\sqrt{1-V^2/c^2}}=\frac{M}{2m}.
$$
The <proper time> along its <world line> satisfies $d\tau=dt/\gamma_A$. Since the first decay occurs at the spacetime origin, integration gives
$$
\boxed{t=\frac{M}{2m}\tau,\qquad x=-\frac{MV}{2m}\tau.}
$$
Equivalently these are the components of A's constant <four-velocity> times $\tau$. The assumption $m>0$ is needed for a daughter rest frame and its proper-time clock; a massless daughter has neither.
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