Past exam of the physics 1 paper of the University of Oxford / 2022 / 6

The special relativity expression for momentum is $p=\gamma mv$, with Lorentz factor $\gamma =(1-v^2/c^2{)}^{-1/2}$. Equal speeds imply equal Lorentz factors, so the pion and proton momenta are in the ratio of their rest masses:The common dimensionless product is $\gamma v/c=p_p/(m_{p}c)=400/0.94\simeq 425.53$, giving $\gamma \simeq 425.53$ and $v/c\simeq 0.99999724$.

By time dilation, the laboratory mean lifetime is $\tau =\gamma {\tau }_0$, where ${\tau }_0=2.6\times 10^{-8}\mathrm{s}$. The exponential decay law evaluated at travel time $d/v$ gives the surviving pion fractionUse the exact product $\gamma v=(400/0.94)c$ from part (a), rather than approximating the speed separately. With $c=2.99792458\times 10^{8}m/s$,Thus about $88.6\%$ emerge without decaying, and about $11.4\%$ decay inside the pipe. This assumes the given common longitudinal speed and neglects losses from other interactions.