The boy on the tower of height h = 20 m in the figure below throws a ball a horizontal distance of x = 56 m. At what speed, in m/s, is the ball thrown?

To find the speed at which the ball is thrown, we need to use the concept of projectile motion. In projectile motion, the horizontal and vertical motions of an object are independent of each other.

Given:
Height of the tower (h) = 20 m
Horizontal distance traveled (x) = 56 m

The horizontal and vertical motions of the ball can be analyzed separately.

Horizontal Motion:
The horizontal motion of the ball is at a constant speed as there is no external force acting on it horizontally. Therefore, we can use the formula for speed (v) = distance (x) / time (t).

Using the horizontal distance traveled (x = 56 m), we assume that the ball was in the air for the same amount of time as it took to reach its maximum height and fall back down. This is because the time it takes for the ball to reach its maximum height is the same as the time it takes to fall back down to the ground (neglecting air resistance).

So, to find the time of flight (t) of the ball, we can use the equation for the time it takes to fall from a certain height:

h = (1/2) * g * t^2

where g is the acceleration due to gravity (approximately 9.8 m/s^2).

Plugging in the height of the tower (h = 20 m) into the equation and solving for t:

20 = (1/2) * 9.8 * t^2
t^2 = (20 * 2) / 9.8
t^2 = 4.08
t ≈ 2.02 s

Since the ball takes t = 2.02 s to travel a horizontal distance of x = 56 m, we can now find the horizontal speed (v) using the equation:

v = x / t = 56 m / 2.02 s ≈ 27.72 m/s

Therefore, the ball is thrown with a speed of approximately 27.72 m/s.

To find the speed at which the ball is thrown, we can use the horizontal distance it traveled and the height of the tower.

Here's how you can calculate it step-by-step:

Step 1: Assume the ball was thrown horizontally.

Step 2: Use the equation for horizontal distance:
x = v * t,
where v is the initial horizontal velocity and t is the time of flight.

Step 3: Since the motion in the vertical direction is influenced by gravity, we can use the equation:
h = (1/2) * g * t^2,
where g is the acceleration due to gravity (approximately 9.8 m/s^2).

Step 4: Rearrange the equation for t:
t = sqrt((2 * h) / g).

Step 5: Substitute the value of t in the equation for x:
x = v * sqrt((2 * h) / g).

Step 6: Rearrange the equation for v:
v = x / sqrt((2 * h) / g).

Step 7: Plug in the given values:
h = 20 m,
x = 56 m,
g = 9.8 m/s^2.

Step 8: Calculate the value of v:
v = 56 / sqrt((2 * 20) / 9.8).

Step 9: Calculate v to find the speed at which the ball is thrown.