Two small balls, each of mass m are placed on  a smooth table, connected with a light string of  length 2l, as shown in the figure. The midpoint of  the string is pulled along y direction by applying a constant force F. Find the relative speed of the 
two particles when they are about to collide. If the  two masses collide and stick to each other, how  much kinetic energy is lost. 

Position of the balls at a later time has been shown 

$2\mathrm{Tcos}\mathrm{\theta }=\mathrm{F}\Rightarrow \mathrm{T}=\frac{\mathrm{F}}{2\cos \mathrm{\theta }}$
$$\begin{gathered} \mathrm{x}=\mathrm{lsin}\mathrm{\theta } \\ \mathrm{dx}=1\cos \mathrm{\theta d\theta } \end{gathered}$$
For motion of a ball in x direction 
 

$$\begin{gathered} \mathrm{Tsin}\mathrm{\theta }=\mathrm{m}\frac{{\mathrm{dv}}_{\mathrm{x}}}{\mathrm{dt}}\Rightarrow {\mathrm{v}}_{\mathrm{x}}\frac{{\mathrm{dv}}_{\mathrm{x}}}{\mathrm{dx}}=\frac{\mathrm{T}}{\mathrm{m}}\sin \mathrm{\theta } \\ \Rightarrow {\mathrm{v}}_{\mathrm{x}}{\mathrm{dv}}_{\mathrm{x}}=\frac{\mathrm{T}}{\mathrm{m}}\sin \mathrm{\theta lcos}\mathrm{\theta d\theta }\left[\text{ using (ii) }\right] \\ =\frac{\mathrm{Fl}}{2\mathrm{m}}\sin \mathrm{\theta d\theta }\left[\text{ using (i) }\right] \\ \therefore {\int }_0^{{\mathrm{v}}_{\mathrm{x}}} {\mathrm{v}}_{\mathrm{x}}{\mathrm{dv}}_{\mathrm{x}}=-\frac{\mathrm{Fl}}{2\mathrm{m}}{\int }_{\mathrm{\pi }/2}^0 \sin \mathrm{\theta d\theta } \end{gathered}$$
[we have placed negative sign as $\theta$ is decreasing] 
$\frac{{\mathrm{v}}_{\mathrm{x}}^2}{2}=\frac{\mathrm{F\ell }}{2\mathrm{m}}\Rightarrow {\mathrm{v}}_{\mathrm{x}}=\sqrt{\frac{\mathrm{Fl}}{\mathrm{m}}}$
Relative velocity of approach before collision 
$=2{\mathrm{v}}_{\mathrm{x}}=2\sqrt{\frac{\mathrm{Fl}}{\mathrm{m}}}$ 
[The two balls will always have zero relative velocity in y direction]
It follows from momentum conservation that after collision the two balls will lose all their velocity in x direction. 
$\therefore \text{ loss in }\mathrm{KE}=2\times \frac{1}{2}{\mathrm{mv}}_{\mathrm{x}}^2=\mathrm{Fl}$