Analysis:
To find the average speed of the ball over its entire trajectory, we need to consider the motion of the ball in two parts: the upward motion and the downward motion.

1. Upward motion:
During the upward motion, the ball experiences a constant acceleration due to gravity in the opposite direction to its velocity. The initial velocity is u and the final velocity is 0 when the ball reaches its maximum height.

Using the equation of motion v = u + at, where v is the final velocity, a is the acceleration, and t is the time, we can find the time taken for the upward motion as follows:
0 = u - gt
t = u/g

The maximum height reached by the ball can be found using the equation of motion s = ut + (1/2)at^2, where s is the displacement. Substituting the values, we get:
s = ut - (1/2)gt^2
s = u(u/g) - (1/2)g(u/g)^2
s = u^2/2g

2. Downward motion:
During the downward motion, the ball experiences a constant acceleration due to gravity in the same direction as its velocity. The initial velocity is 0 and the final velocity is v when the ball hits the ground.

Using the equation of motion v = u + at, we can find the time taken for the downward motion as follows:
v = 0 + gt
t = v/g

The displacement during the downward motion can be found using the equation of motion s = ut + (1/2)at^2:
s = 0(v/g) + (1/2)g(v/g)^2
s = v^2/2g

Average speed:
The average speed is defined as the total distance traveled divided by the total time taken. In this case, the total distance is twice the maximum height (up and down), and the total time is the sum of the time taken for the upward and downward motions.

Total distance = 2 * (u^2/2g)
Total time = (u/g) + (v/g)

Average speed = Total distance / Total time
Average speed = (2 * (u^2/2g)) / ((u/g) + (v/g))
Average speed = (u^2/v + u) / g

Conclusion:
The average speed of the ball over its entire trajectory is given by (u^2/v + u) / g, where u is the initial speed, v is the speed at which the ball hits the ground, and g is the acceleration due to gravity.