Question

Two equal discs are in contact on a table. A third disc of same mass but of double radius strikes them symmetrically and remains at rest after impact. The co-efficient of restitution is:

• A. $\frac{2}{3}$
• B. $\frac{1}{3}$
• C. $\frac{9}{16}$
• D. $\frac{\sqrt{3}}{2}$

Answer 1

The correct option is C $\frac{9}{16}$


Let $u$ be initial velocity of big disc and $\text{v}$ be velocity of small disc after impact
In $\Delta AOB$,
$AB=\sqrt{(AO{)}^2-(OB{)}^2}$
$=\sqrt{9R^2-R^2}$
$AB=2\sqrt{2}R$
Now $\text{cos}\theta =\frac{AB}{AO}=\frac{2\sqrt{2}R}{3R}=\frac{2\sqrt{2}}{3}$
Applying momentum conservation,
$mu=\text{mv cos}\theta +\text{mv cos}\theta$
$u=2\text{v cos}\theta$
Now $e=\frac{\text{separation velocity along line of impact}}{\text{velocity of approcah along line of impact}}$
$\Rightarrow e=\frac{\text{v}}{\text{u cos}\theta }=\frac{1}{2{\text{cos}}^2\theta }$
$=\frac{1}{2\times {\left(\frac{2\sqrt{2}}{3}\right)}^2}=\frac{9}{16}$
$\therefore e=\frac{9}{16}$
The correct option is (C).