A sphere of mass m moving with velocity v col | vedclass

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Question

(C) Let the mass of both spheres be $m$. The initial velocity of the first sphere is $v$ and the second sphere is $0$. Let their final velocities be $

Vedclass · Accepted Answer

$\frac{e+1}{2}$

Vedclass · Answer

(C) Let the mass of both spheres be $m$. The initial velocity of the first sphere is $v$ and the second sphere is $0$. Let their final velocities be $V_1$ and $V_2$ respectively.By the law of conservation of linear momentum:$m v + m(0) = m V_1 + m V_2$$v = V_1 + V_2$  --- $(1)$By the definition of the coefficient of restitution $e$:$e = \frac{V_2 - V_1}{u_1 - u_2}$$e = \frac{V_2 - V_1}{v - 0}$$e v = V_2 - V_1$ --- $(2)$Adding equations $(1)$ and $(2)$:$v + e v = (V_1 + V_2) + (V_2 - V_1)$$v(1 + e) = 2 V_2$$V_2 = \frac{v(e + 1)}{2}$The ratio of the final velocity of the second sphere $(V_2)$ to the initial velocity of the first sphere $(v)$ is:$\frac{V_2}{v} = \frac{e + 1}{2}$

$A$ sphere of mass $m$ moving with velocity $v$ collides head-on with another sphere of the same mass which is at rest. The ratio of the final velocity of the second sphere to the initial velocity of the first sphere is (where $e$ is the coefficient of restitution and the collision is inelastic).

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