Question

A man standing on the edge of a cliff at some height above the ground throws ball 1 straight up with intial speed u and then throws ball 2 straight down with the same intial speed then what happens

Answer 1

If we neglect the effect of air-resistance, the speed of the ball, when it hits the ground for both the cases will be same.
When the ball thrown upward with initial velocity v₀ , return from its upward journey and crosses the point corresponding to observer position (from where it was thrown upwards), the speed of the ball will be equal to the initial speed. Therefore the motion of the ball resembles the situation in which the ball was thrown downward with initial speed v₀.
Mathematically, one can see that the time taken by the ball (for situation in which it was thrown upward) to reach the highest point is equal to the time taken by the ball to reach cross the same point again during its downfall. Therefore, under constant acceleration, the magnitude of the change in velocity in the upward motion will be equal to the change in velocity in the downward motion. However this is true only when the effect of air resistance is neglected.
Therefore if the object loses its velocity in reaching the highest point, it will gain the velocity when it crosses the point from which it was released.
Keeping that in mind, it is clear that the final speed of the ball for both the situation would be same. Hence the final speed in either of the situation will exactly be equal.