Velocity of A relative to B is given by
$\overrightarrow {{v_{A/B}}} = \overrightarrow {{v_A}} - \overrightarrow {{v_B}} = \overrightarrow v - \overrightarrow {{v_1}}$    ...(i)

By taking x-components of equation (i), we get
$0 = v\sin \theta - {v_1} \Rightarrow \sin \theta = {{{v_1}} \over v}$    ...(ii)

By taking Y-components of equation (i), we get
${v_y} = v\cos \theta$    ...(iii)

Time taken by boy at A to catch the boy at B is given by

t = ${{{\mathop{\rm Relative}\nolimits} \,displacement\,along\,Y - axis} \over {{\mathop{\rm Relative}\nolimits} \,velocity\,along\,Y - axis}}$

= ${a \over {v\cos \theta }} = {a \over {v.\sqrt {1 - {{\sin }^2}\theta } }} = {a \over {v.\sqrt {1 - {{\left( {{{{v_1}} \over v}} \right)}^2}} }}$
[From equation (i)]

= ${a \over {v.\sqrt {{{{v^2} - v_1^2} \over {{v^2}}}} }} = {a \over {\sqrt {{v^2} - v_1^2} }}$

= $\sqrt {{{{a^2}} \over {{v^2} - v_1^2}}}$