Question 303: Centre of Mass and Rotational Motion

Question

A sphere of mass ' $m$ ', moving with velocity ' $3 u$ ' collides head-on with another identical sphere at rest. If ' $e$ ' is coefficient of restitution then what will be the ratio of velocity of the second sphere to that of first sphere after collision?

Accepted Answer

$$ \begin{aligned} &	ext { Coefficient of restitution: }\ &\begin{array}{ll}  & \mathrm{e}=\frac{\mathrm{v}_2-\mathrm{v}_1}{\mathrm{u}_1-\mathrm{u}_2} \ & 	ext { Given: } \mathrm{u}_1=3 \mathrm{u}, \mathrm{u}_2=0 \ 	herefore \quad & \mathrm{e}=\frac{\mathrm{v}_2-\mathrm{v}_1}{3 \mathrm{u}}=\frac{\mathrm{v}_2}{3 \mathrm{u}}-\frac{\mathrm{v}_1}{3 \mathrm{u}} \end{array} \end{aligned} $$
$$ v_2-v_1=3 u e \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(i)$$
By law of conservation of momentum,
$$ \begin{aligned} & \mathrm{mu}_1+\mathrm{mu}_2=\mathrm{mv}_1+\mathrm{mv}_2 \ & 3 \mathrm{mu}=\mathrm{mv}_1+\mathrm{mv}_2 \ & \mathrm{v}_1+\mathrm{v}_2=3 \mathrm{u} \ & \mathrm{v}_1=3 \mathrm{u}-\mathrm{v}_2 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(ii)\end{aligned} $$
From equation (i) and (ii),
$$ \begin{aligned} & v_2-\left(3 u-v_2ight)=3 u e \Rightarrow 2 v_2-3 u=3 u e \ & v_2=\frac{3 u(1+e)}{2} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(iii)\end{aligned} $$
For $\mathrm{v}_1$
$$ \begin{aligned} & v_1=3 u-v_2=3 u-\frac{3 u(1+e)}{2} \ & v_1=\frac{3 u(1-e)}{2} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(iv)\end{aligned} $$
From equation (iii) and (iv),
$$ 	ext { Ratio }=\frac{v_2}{v_1}=\frac{\frac{3 u(1+e)}{2}}{\frac{3 u(1-e)}{2}}=\frac{1+e}{1-e} $$

Answer

$\frac{1-\mathrm{e}}{1+\mathrm{e}}$

Answer

$\frac{1+\mathrm{e}}{1-\mathrm{e}}$

Answer

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

Answer

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

Centre of Mass and Rotational Motion

A sphere of mass ' m ', moving with velocity ' 3 u ' collides head-on with another identical sphere at rest. If ' e ' is coefficient of restitution then what will be the ratio of velocity of the second sphere to that of first sphere after collision?

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