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  4. A ball of mass m moving with a constant velocity u strikes against a ball of same mass at rest. If e = coefficient of restitution, then what will be the ratio of velocity of two balls after collision ?

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Q. A ball of mass $m$ moving with a constant velocity $u$ strikes against a ball of same mass at rest. If $e$ = coefficient of restitution, then what will be the ratio of velocity of two balls after collision ?

JIPMERJIPMER 2014Work, Energy and Power

Solution:

Here, $m_1 = m, m_2 = m, u_1 = u, u_2 = 0$
Coefficient of restitution, $e = \frac{v_2 - v_1}{u_1 - u_2}$
$e = \frac{v_2 -v_1}{u -0}$
or $v_2 - v_1 = eu$ ....(i)
According to law of conservation of linear momentum,
$m_1 \, u_1 + m_2 \, u_2 = m_1 \, v_1 + m_2 \, v_2$
$\Rightarrow \:\: mu = m (v_1 + v_2)$
or $v_1 + v_2 = u$ ...(ii)
Solving equations (i) and (ii) for $v_1$ and $v_2$, we get
$v_1 = \frac{u(1 -e)}{2} , v_2 = \frac{u(1 + e)}{2}$
$\therefore \:\: \frac{v_1}{v_2} = \frac{1 -e}{1 +e}$