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

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 $v/c\simeq 0.99999724$.