 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/ib/paper-4/18f/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Before its second decay A moves uniformly with

x = - V t

γ_{A} = \frac{1}{1 - V ^{2} / c ^{2}} = \frac{M}{2 m} .

The proper time along its world line satisfies

d τ = d t / γ_{A}

. Since the first decay occurs at the spacetime origin, integration gives

t = \frac{M}{2 m} τ, x = - \frac{M V}{2 m} τ .

Equivalently these are the components of A's constant four-velocity times

τ

. 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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