A bob of mass m, suspended by a string of length ${\mathcal{l}}_1$, is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass m suspended by a string of length ${\mathcal{l}}_2$, which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio  $\frac{{\mathrm{\ell }}_1}{{\mathrm{\ell }}_2}$  is