A sphere of mass $m$ , moving with velocity $u$ , collides head on with an identical sphere at rest. If the collision is inelastic with coefficient of restitution $e$ , then, the ratio of their speeds after collision will be :

e

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\frac{2 e}{1 + e}

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\frac{1 - e}{1 + e}

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\frac{1 + e}{1 - e}

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Solution

The correct option is C $\frac{1 - e}{1 + e}$
The law of conservation of linear momentum tells us that the overall momentum before the collision must be equal to the overall momentum after a collision.
Since the spheres have identical masses, we can write
$m u + m \times 0 = m v_{A} + m v_{B}$
$u = v_{A} + v_{B}$
From the definition of the coefficient of restitution, we know that
$e = \frac{v_{B} - v_{A}}{u}$
solving above two equations
$e \times (v_{A} + v_{B}) = v_{B} - v_{A}$
$v_{B} (1 - e) = v_{A} (1 + e)$
$\frac{v_{A}}{v_{B}} = \frac{1 - e}{1 + e}$

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A sphere of mass m, moving with velocity u, collides head on with an identical sphere at rest. If the collision is inelastic with coefficient of restitution e, then, the ratio of their speeds after collision will be :

A sphere of mass $m$ , moving with velocity $u$ , collides head on with an identical sphere at rest. If the collision is inelastic with coefficient of restitution $e$ , then, the ratio of their speeds after collision will be :