A ball is projected horizontally from the top of a building and another is dropped with zero velocity from the same point at the same time, which one will hit the ground first?

Both the balls will hit the ground at the same time. A ball projected horizontally has no vertical component of velocity. Whatever be the horizontal component of velocity, it has no effect on the vertical motion of the ball. Since only the force of gravity acts on both the cases, the time required to hit the ground depends only on the height of fall and acceleration due to gravity. If $t$ is the time required to hit the ground then $t=\sqrt{\frac{2H}{g}}$ for both of them.

[Read: Projectile Fired Horizontally from the Top of a Tower]