A sphere of mass m moving with a constant velocity u hits another stationary sphere of the same mass. If e is the coefficient of restitution, then ratio of velocities of the first sphere and second sphere after collision will be:

Engineering

Physics

Collision

Question

$(\frac{1 - e}{1 + e})$

$(\frac{1 + e}{1 - e})$

$(\frac{e - 1}{e + 1})$

$(\frac{e + 1}{e + 1})$

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For two spheres of equal mass m, with first moving at velocity u and second stationary, we apply conservation of momentum and the coefficient of restitution (e) definition.

Momentum conservation: $m v_{1} + m v_{2} = m u$

Restitution: $e = \frac{v_{2} - v_{1}}{u}$

Solving these equations gives the velocities after collision: $v_{1} = \frac{u (1 - e)}{2}$ and $v_{2} = \frac{u (1 + e)}{2}$

The ratio of first to second sphere's velocity is therefore $\frac{v_{1}}{v_{2}} = \frac{1 - e}{1 + e}$

Final Answer: $(\frac{1 - e}{1 + e})$