Question
A sphere collides with another stationary sphere of equal mass. If the coefficient of resitution is $e$ then what will be the ratio of velocities of the spheres after the collision?
✓
Answer
Let the mass of both the spheres is $m$ and the initial velocity of first sphere is $u$. Afte the collision the velocities of first and second spheres are $v_1$ and $v_2$ respectively. Second sphere is at rest initially.By the low of conservation of momentum,
$\begin{aligned}
m u & =m v_1+m v_2 \\
v_1+v_2 & =u.....(1)
\end{aligned}$
Restitution coefficient
$\begin{aligned}
e & =-\left(\frac{v_1-v_2}{u}\right) \\
v_1-v_2 & =-e u......(2)
\end{aligned}$
By addition of equation (1) and (2)
$\begin{aligned}
2 v_1 & =u-e u=(1-e) u \\
v_1 & =\frac{1-e}{2} u......(3)
\end{aligned}$
By substracting the equation (2) from (1)
$\begin{aligned}
2 v_2 & =u+e u \\
2 v_2 & =(1+e) u \\
\Rightarrow \quad v_2 & =\left(\frac{1+e}{2}\right) u.....(4)
\end{aligned}$
From equtions (3) and (4)
$\frac{v_1}{v_2}=\frac{1-e}{1+e}$
$\begin{aligned}
m u & =m v_1+m v_2 \\
v_1+v_2 & =u.....(1)
\end{aligned}$
Restitution coefficient
$\begin{aligned}
e & =-\left(\frac{v_1-v_2}{u}\right) \\
v_1-v_2 & =-e u......(2)
\end{aligned}$
By addition of equation (1) and (2)
$\begin{aligned}
2 v_1 & =u-e u=(1-e) u \\
v_1 & =\frac{1-e}{2} u......(3)
\end{aligned}$
By substracting the equation (2) from (1)
$\begin{aligned}
2 v_2 & =u+e u \\
2 v_2 & =(1+e) u \\
\Rightarrow \quad v_2 & =\left(\frac{1+e}{2}\right) u.....(4)
\end{aligned}$
From equtions (3) and (4)
$\frac{v_1}{v_2}=\frac{1-e}{1+e}$
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