A steel ball is dropped from a building's roof and passes a window, taking 0.117 s to fall from the top to the bottom of the window, a distance of 1.22 m. It then falls to a sidewalk and bounces back past the window, moving from bottom to top in 0.117 s. Assume that the upward flight is an exact reverse of the fall. The time spent below the bottom of the window is 1.90 s. How tall is the building?

Question

A steel ball is dropped from a building's roof and passes a window, taking 0.117 s to fall from the top to the bottom of the window, a distance of 1.22 m. It then falls to a sidewalk and bounces back past the window, moving from bottom to top in 0.117 s. Assume that the upward flight is an exact reverse of the fall. The time spent below the bottom of the window is 1.90 s. How tall is the building?

Answer 1

To find the height of the building, we need to determine two things: the time it takes for the ball to reach the window on its initial fall and the total time the ball spends in the air during the full cycle.

Let's start by finding the time it takes for the ball to reach the window on its initial fall.

Given:
- Time taken for the ball to fall from the top to the bottom of the window: 0.117 s
- Distance covered during this time: 1.22 m

To find the time it takes to reach the window, we can use the equation for displacement:

s = ut + 1/2at^2

Since the ball starts from rest (u = 0) and the only force acting on it is due to gravity (a = 9.8 m/s²), the equation simplifies to:

s = 1/2at^2

Substituting the known values:

1.22 = 0.5 * 9.8 * t^2

Solving for t:

1.22 = 4.9t^2

t^2 = 1.22/4.9

t^2 = 0.249

t = √0.249

t ≈ 0.499 s

So, it takes approximately 0.499 seconds for the ball to reach the window on its initial fall.

Now, let's calculate the total time the ball spends in the air during the full cycle.

Given:
- Time spent below the bottom of the window: 1.90 s

The ball spends the same amount of time on its initial fall from the top to the bottom of the window as it does during the entire period below the window. Therefore, the total time in the air is:

Total time in the air = Time for initial fall + Time below the window

Total time in the air = 0.499 + 1.90

Total time in the air ≈ 2.399 seconds

Finally, to find the height of the building, we can use the equation for motion:

s = ut + 1/2at^2

Where:
- s is the displacement (height of the building)
- u is the initial velocity (0 since the ball starts from rest)
- a is acceleration (due to gravity, a = 9.8 m/s²)
- t is the total time in the air (2.399 s)

Simplifying the equation:

s = 0 + 0.5 * 9.8 * (2.399)^2

s = 0.5 * 9.8 * 5.754

s ≈ 28.175 m

Therefore, the height of the building is approximately 28.175 meters.