A ball P collides with another identical ball | vedclass

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(A) Let the mass of both balls be $m$. Let the initial velocity of ball $P$ be $u_1$ and ball $Q$ be $u_2 = 0$. After the collision,let their velociti

Vedclass · Accepted Answer

$1/3$

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(A) Let the mass of both balls be $m$. Let the initial velocity of ball $P$ be $u_1$ and ball $Q$ be $u_2 = 0$. After the collision,let their velocities be $v_1$ and $v_2$ respectively.From the conservation of linear momentum: $m u_1 + 0 = m v_1 + m v_2 \implies u_1 = v_1 + v_2$.From the definition of the coefficient of restitution $e$: $e = \frac{v_2 - v_1}{u_1 - u_2} \implies e u_1 = v_2 - v_1$.Adding the two equations: $(1+e) u_1 = 2 v_2 \implies v_2 = \frac{1+e}{2} u_1$.Subtracting the equations: $(1-e) u_1 = 2 v_1 \implies v_1 = \frac{1-e}{2} u_1$.Given that $v_2 = 2 v_1$,we substitute the expressions:$\frac{1+e}{2} u_1 = 2 \left( \frac{1-e}{2} \right) u_1$.$1 + e = 2(1 - e) \implies 1 + e = 2 - 2e$.$3e = 1 \implies e = 1/3$.

$A$ ball $P$ collides with another identical ball $Q$ at rest. For what value of the coefficient of restitution $e$ will the velocity of ball $Q$ become two times that of ball $P$ after the collision?

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