Question

Two small bids each of mass m is attached to smooth ring of mass m as shown in the figure. The whole system is kept on smooth horizontal table. Initially each bid has given velocity $v_0$ as shown in figure. Now if velocity of bid just before impact $\frac{\sqrt{x}{v}_0}{3}$. Find x. ___

Answer 1

By conservation of linear momentum $2m{v}_0=3m{v}_x$
$v_x=\frac{2v_0}{3}$
By energy conservation
$\frac{1}{2}m{v}_0^2\times 2=\frac{1}{2}m(v_x^2+v_y^2)\times 2+\frac{1}{2}m{v}_x^2$
$2v_0^2=2(v_x^2+v_y^2)+v_x^2$
$2v_0^2=3\times \frac{4v_0^2}{9}+2v_y^2$
$\frac{2v_0^2}{3}=2v_y^2\Rightarrow v_y=\frac{v_0}{\sqrt{3}}$
$v_{bid}=\sqrt{v_x^2+v_y^2}=\sqrt{\frac{4v_0^2}{9}}=\sqrt{\frac{7v_0^2}{9}}=\frac{\sqrt{7}{v}_0}{3}$