The boy on the tower of height h = 10 m throws a ball a distance of x = 68 m, as shown. At what speed, in m/s, is the ball thrown?

I do not see the figure. If the ball is thrown horizontally, use equations of motion

u = initial vertical velocity = 0 if the ball was thrown horizontally
S = distance travelled, metres
t = time, seconds
g = acceleration due to gravity = 9.81 m/s²

S = ut + (1/2)gt²

Solve for time t.

At the same time, the ball has travelled x=68m at a uniform horizontal velocity of vh.
So vh=S/t

If the ball is not thrown horizontally, we need to see a description of the figure.

To find the speed at which the ball is thrown, we need to use the formula for the velocity of an object in free fall. We can break down this problem into two parts: the horizontal motion and the vertical motion.

1. Vertical Motion:
The height of the tower (h) is given as 10 m. We know that the ball was thrown vertically upward, so it reaches its maximum height at some point before falling back down. The ball's final velocity at the maximum height is 0 m/s because it momentarily stops there before accelerating downward due to gravity.

2. Horizontal Motion:
The distance traveled in the horizontal direction (x) is given as 68 m. Since there is no horizontal force acting on the ball during its flight, the horizontal component of velocity remains constant throughout the motion.

Now, let's calculate the initial vertical velocity:

1. Use the equation of motion for vertical motion:
vf^2 = vi^2 + 2gh

Here, vf is the final vertical velocity (0 m/s), vi is the initial vertical velocity, g is the acceleration due to gravity (-9.8 m/s^2), and h is the height of the tower (10 m).

Putting the values into the equation:
(0 m/s)^2 = vi^2 + 2 * (-9.8 m/s^2) * 10 m
0 = vi^2 - 196 m^2/s^2
vi^2 = 196 m^2/s^2

Taking the square root of both sides:
vi = 14 m/s (approximately)

2. Now, we can find the speed at which the ball is thrown by calculating the magnitude of the initial velocity vector. Since the ball is thrown vertically, the initial speed and initial velocity in the horizontal direction are the same.

Therefore, the speed at which the ball is thrown is approximately 14 m/s.

To summarize, the ball is thrown with a speed of approximately 14 m/s.