\[(a)\;\frac{a_1^2}{a_2}\quad (b)\;\frac{a_1+a_2}{2} \quad (c)\;\frac{{a_2}^2}{a_1} \quad (d)\sqrt {a_1a_2}\]

Let s be the distance travelled by each car

$v_A=\sqrt {2 a_1 s}$

Also $v_B=\sqrt {2 a_2 s}$

$t_1=\sqrt {\large\frac{2s}{a_1}}$

$t_2=\sqrt {\large\frac{2s}{a_2}}$

$\sqrt {2 a_1 s}-\sqrt {2 a_2 s}=v_0$

and

$\sqrt {\large\frac{2s}{a_2}}-\sqrt {\large\frac{2s}{a_1}}=t_0$

$\large\frac{v_0}{t_0}=\frac{\sqrt {a_1}-\sqrt {a_2}}{\Large\frac{1}{\sqrt{a_2}}-\frac{1}{\sqrt{a_1}}}$$=\sqrt{a_1a_2}$

Hence d is the correct answer.

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