Coefficient of restitution

Question

Two spheres moving in opposite directions with velocities ${u}_{1}$ and ${u}_{2}$ collide will each other. If e is the coefficient of restitution, their distance of separation at time ‘t’ after collision is

Easy

Solution

$\frac{{v}_{2}-{v}_{1}}{{u}_{1}-{u}_{2}}=e\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\frac{vel.\text{\hspace{0.17em}}of\text{\hspace{0.17em}}separation}{vel.\text{\hspace{0.17em}}of\text{\hspace{0.17em}}approach}$along the line of impact

$({v}_{2}-{v}_{1})=e({u}_{1}-{u}_{2})$

Separation after time t, $({v}_{2}-{v}_{1})t=e({v}_{1}-{v}_{2})t$

As ${v}_{1}and{v}_{2}$ are opposite direction

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