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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 the ratio of the velocity of two spheres after collision will be
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Verified Answer
The correct answer is:
$\frac{1-e}{1+e}$
Step 1: Apply momentum conservation
Since, there is no external force acting on the system, therefore momentum is conserved.
Let $\mathrm{v}_1$ and $\mathrm{v}_2$ be the velocities after collision
$\begin{aligned}
& \mathrm{P}_{\mathrm{i}}=\mathrm{P}_{\mathrm{f}} \\
& \Rightarrow \mathrm{mu}+\mathrm{o}=\mathrm{mv}_1+\mathrm{mv}_2 \\
& \Rightarrow \mathrm{u}=\mathrm{v}_1+\mathrm{v}_2 \ldots \text { (1) }
\end{aligned}$
Step 2: Coefficient of restitution (e)
Coefficient of restitution is defined as:
$\begin{aligned}
& e=\frac{\text { Relative velocity of separation(after collision) }}{\text { Relative velocity of approach(before collision) }}=\frac{v_2-v_1}{u_1-u_2} \\
& \Rightarrow e=\frac{\mathrm{v}_2-v_1}{\mathrm{u}-\mathrm{o}} \Rightarrow \mathrm{v}_2-\mathrm{v}_1=\mathrm{eu} \ldots . \text { (2) }
\end{aligned}$
Step 3: Solving above equations
Adding eq (1) and $(2) \Rightarrow 2 \mathrm{v}_2=(1+e) \mathrm{u} \Rightarrow \mathrm{v}_2=\frac{(1+e)}{2} \mathrm{u} \ldots$
Subtracting eq (2) from (1) $\Rightarrow 2 v_1=(1-e) u \Rightarrow v_1=\frac{(1-e)}{2}$
Dividing eq (4) by (3) $\Rightarrow \frac{\mathrm{v}_1}{\mathrm{v}_2}=\frac{1-\mathrm{e}}{1+\mathrm{e}}$
Since, there is no external force acting on the system, therefore momentum is conserved.
Let $\mathrm{v}_1$ and $\mathrm{v}_2$ be the velocities after collision
$\begin{aligned}
& \mathrm{P}_{\mathrm{i}}=\mathrm{P}_{\mathrm{f}} \\
& \Rightarrow \mathrm{mu}+\mathrm{o}=\mathrm{mv}_1+\mathrm{mv}_2 \\
& \Rightarrow \mathrm{u}=\mathrm{v}_1+\mathrm{v}_2 \ldots \text { (1) }
\end{aligned}$
Step 2: Coefficient of restitution (e)
Coefficient of restitution is defined as:
$\begin{aligned}
& e=\frac{\text { Relative velocity of separation(after collision) }}{\text { Relative velocity of approach(before collision) }}=\frac{v_2-v_1}{u_1-u_2} \\
& \Rightarrow e=\frac{\mathrm{v}_2-v_1}{\mathrm{u}-\mathrm{o}} \Rightarrow \mathrm{v}_2-\mathrm{v}_1=\mathrm{eu} \ldots . \text { (2) }
\end{aligned}$
Step 3: Solving above equations
Adding eq (1) and $(2) \Rightarrow 2 \mathrm{v}_2=(1+e) \mathrm{u} \Rightarrow \mathrm{v}_2=\frac{(1+e)}{2} \mathrm{u} \ldots$
Subtracting eq (2) from (1) $\Rightarrow 2 v_1=(1-e) u \Rightarrow v_1=\frac{(1-e)}{2}$
Dividing eq (4) by (3) $\Rightarrow \frac{\mathrm{v}_1}{\mathrm{v}_2}=\frac{1-\mathrm{e}}{1+\mathrm{e}}$
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