A sphere of mass $m$ moving with a uniform velocity $v$ hits another identical sphere at rest. If the coefficient of restitution is $e$ , the ratio of the velocities of the two spheres after the collision is $\frac{e - 1}{e + 1}$ .

True

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False

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Solution

The correct option is B False
After collision they move with velocities
$V_{1}$ & $V_{2}$
By conservation of momentum
mV=MV_1+mV_2\\ \therefore V=V_1+V_2\rightarrow(1)$
Given that co-efficient of restitution $= e ∴ e = \frac{V_{2} - V_{1}}{V - O} \to e V = V_{2} - V_{1} \to (2)$
$(1)$ in $(2) \Rightarrow e (V_{1} + V_{2}) = V_{2} - V_{1} \Rightarrow e V_{1} + V_{1} = V_{2} - e V_{2} \Rightarrow \frac{V_{1}}{V_{2}} = \frac{1 - e}{1 + e}$

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A sphere of mass m moving with a uniform velocity v hits another identical sphere at rest. If the coefficient of restitution is e, the ratio of the velocities of the two spheres after the collision is e−1e+1.

The correct option is B FalseAfter collision they move with velocitiesV1 & V2By conservation of momentummV=MV_1+mV_2\\ \therefore V=V_1+V_2\rightarrow(1)$Given that co-efficient of restitution=e∴e=V2−V1V−O→eV=V2−V1→(2)(1) in (2)⇒e(V1+V2)=V2−V1⇒eV1+V1=V2−eV2⇒V1V2=1−e1+e

A sphere of mass $m$ moving with a uniform velocity $v$ hits another identical sphere at rest. If the coefficient of restitution is $e$ , the ratio of the velocities of the two spheres after the collision is $\frac{e - 1}{e + 1}$ .