Past exam of the physics A course of the University of Cambridge / 2021 / Paper 1 / Section C / C8 / d / Solution

For an independent count with a Poisson distribution, collected for exposure $T_i$ at rate $R_i$, the expected count is $N_i=R_i{T}_i$ and its standard deviation is $\sqrt{N_i}$. The fractional rate uncertainty is consequentlyA target of $0.15$ requires $N_i\simeq (0.15{)}^{-2}=44.44$ counts at each proper-time setting. The exposure time for a Poisson rate at fixed fractional precision is therefore $T_i=1/(0.15^2{R}_i)$, givingSince the settings are acquired independently rather than simultaneously,Nearly three quarters of the exposure is spent at the latest, lowest-rate setting. Proper time in picoseconds labels the decay distribution; it is not the laboratory acquisition time in days. This estimate assumes counting statistics dominate, with no additional background-subtraction uncertainty or acquisition overhead.