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Velocity of Separation and Approach

A smooth sphe...

Question

A smooth sphere of mass $M$ moving with velocity $u$ directly collides elastically with another sphere of mass $m$ at rest. After the collision, their final velocities are $V$ and $v$ respectively. The value of $v$ is:

\frac{2 u M}{m}

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\frac{2 u m}{M}

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\frac{2 u}{1 + \frac{m}{M}}

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\frac{2 u}{1 + \frac{M}{m}}

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Solution

The correct option is C $\frac{2 u}{1 + \frac{m}{M}}$
Final velocity of mass $M$ , $v_{2} = V$
Final velocity of mass $m$ , $v_{1} = v$
We know that,
Coefficient of restitution,
$\frac{v_{2} - v_{1}}{u_{2} - u_{1}} = - e$
Given,
$u_{2} = u, v_{2} = V, u_{1} = 0, v_{1} = v$
$∴ \frac{V - v}{u - 0} = - e$
For elastic collision,
$∴ \frac{V - v}{u - 0} = - 1$
$\begin{matrix} V = v - u \\ (1) \end{matrix}$
Applying conservation of momentum just before and after the collision,
$M u + 0 = M V + m v$
Using euqation $(1)$ ,
$\Rightarrow M u = M (v - u) + m v$
$\Rightarrow v = \frac{2 M u}{M + m} = \frac{2 u}{1 + \frac{m}{M}}$

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Velocity of Separation and Approach

Standard XI Physics

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A smooth sphere of mass M moving with velocity u directly collides elastically with another sphere of mass m at rest. After the collision, their final velocities are V and v respectively. The value of v is:

A smooth sphere of mass $M$ moving with velocity $u$ directly collides elastically with another sphere of mass $m$ at rest. After the collision, their final velocities are $V$ and $v$ respectively. The value of $v$ is: