Water and mercury fill two cylindrical vessels to an identical...

According to Torricelli's law, the velocity of efflux of a liquid from a small hole at a depth $h$ below the free surface is given by $v = \sqrt{2gh}$.

This expression is independent of the density of the liquid.

Since both vessels are filled to an identical height, the depth $h$ is the same for both water and mercury.

Therefore, the velocities of efflux are equal, which gives $v_1 = v_2$.