Vertical strings of same length L support two balls A and B of mass 2m each. There is a small monkey of mass m sitting on ball A. Suddenly, the monkey jumps off the ball A at an angle $\theta$ = 45° to the horizontal and lands exactly on the ball B. Thereafter, the monkey and the ball B just manage to complete the vertical circle.If u be the velocity with which the monkey jumped off the ball A and I be the impulse of the string tension on ball A during the small period when the monkey interacted with the ball to jump off it , then
 

 

(a) Let the monkey jump off A with a velocity having horizontal and vertical components u_x and u_y respectively. A will recoil to left with velocity 
${\mathrm{V}}_{\mathrm{A}}=\frac{{\mathrm{mu}}_{\mathrm{x}}}{2\mathrm{m}}=\frac{{\mathrm{u}}_{\mathrm{x}}}{2}$ 

When the monkey lands on B, it imparts a horizontal velocity to B For (monkey + B) system 
${\mathrm{V}}_{\mathrm{B}}=\frac{{\mathrm{mu}}_{\mathrm{x}}}{3\mathrm{m}}=\frac{{\mathrm{u}}_{\mathrm{x}}}{3}$ [What happens to vertical velocity?]
This velocity is just sufficient for completing the circle
$\therefore \frac{{\mathrm{u}}_{\mathrm{x}}}{3}=\sqrt{5\mathrm{gL}}\Rightarrow {\mathrm{u}}_{\mathrm{x}}=3\sqrt{5\mathrm{gL}}$
$\therefore$Time of flight from A to B for the monkey
$$\begin{gathered} \mathrm{T}=\frac{\mathrm{d}}{{\mathrm{u}}_{\mathrm{x}}}=\frac{\mathrm{d}}{3\sqrt{5\mathrm{gL}}} \\ \therefore \frac{2{\mathrm{u}}_{\mathrm{y}}}{\mathrm{g}}=\frac{\mathrm{d}}{3\sqrt{5\mathrm{gL}}} \\ \therefore {\mathrm{u}}_{\mathrm{y}}=\frac{\mathrm{d}}{6}\sqrt{\frac{\mathrm{g}}{5\mathrm{L}}} \\ \therefore \tan \mathrm{\theta }=\frac{{\mathrm{u}}_{\mathrm{y}}}{{\mathrm{u}}_{\mathrm{x}}}=\frac{\mathrm{d}}{6}\sqrt{\frac{\mathrm{g}}{5\mathrm{L}}}\times \frac{1}{3\sqrt{5\mathrm{gL}}}=\frac{\mathrm{d}}{90}\frac{1}{\mathrm{L}} \\ \text{ If }\mathrm{\theta }={45}^{\circ };\mathrm{d}=90\mathrm{L} \\ \mathrm{u}\text{ can be calculated, }\mathrm{u}=\sqrt{90\mathrm{gL}} \\ \text{ Impulse of tension }={\mathrm{mu}}_{\mathrm{y}}=\mathrm{m}\sqrt{45\mathrm{gL}} \end{gathered}$$