Question

Prove the statement, “When two equal masses undergo one-dimensional elastic collision, they interchange their velocities.” ​

Answer 1

Dear student
let one body is moving along +X with speed u and after collision it moves with speed v and second particle is initially at rest and after collision its speed is v₁
$$\begin{gathered} u\sin gconservationofmomentum \\ mu+0=mv+m{v}_1 \\ u=v+v_1.....1) \\ nowu\sin gconservationofkinecticenergyofsystem\sin cecollisioniselastic \\ \frac{1}{2}m{u}^2+0=\frac{1}{2}m{v}^2+\frac{1}{2}m{v_1}^2 \\ {u}^2=v^2+{v_1}^2.......2) \\ puttingthevalueofeq1ineq2weget \\ {\left(v+v_1\right)}^2=v^2+{v_1}^2 \\ 2v{v}_1=0 \\ henceeitherv=0or{v}_1=0 \\ Nowputtingv=0ineq1weget{v}_1=u \\ andif{v}_1=0wegetv=0 \\ henceitprovesthataftercollisionvelocitygetinterchange \end{gathered}$$