A steel ball is dropped from a building's roof and passes a window, taking 0.13 s to fall from the top to the bottom of the window, a distance of 1.20 m. It then falls to a sidewalk and bounces back past the window, moving from bottom to top in 0.13 s. Assume that the upward flight is an exact reverse of the fall. The time the ball spends below the bottom of the window is 2.13 s. How tall is the building?

To solve this problem, we can break it down into different parts and calculate the distances for each part separately.

Let's call the height of the building H.

First, let's calculate the distance the ball falls from the top to the bottom of the window. We're given that it takes 0.13 seconds to cover a distance of 1.20 meters. Using a basic kinematic equation, we can use the formula:

distance = (initial velocity * time) + (0.5 * acceleration * time^2)

Since the ball is only falling due to gravity, we can assume the acceleration is -9.8 m/s^2 (negative because it's going down). The initial velocity in this case is 0 since the ball is dropped. Rearranging the equation, we get:

1.20 = 0.5 * (-9.8) * (0.13)^2

Solving this equation, we find that the initial fall from the top to the bottom of the window is approximately 1.29 meters.

Next, let's calculate the distance the ball falls from the bottom of the window to the sidewalk. We're given that the ball spends a total time of 2.13 seconds below the bottom of the window. Since the upward flight is an exact reverse of the fall, we can assume the time spent falling from the bottom of the window to the sidewalk is also 2.13 seconds. Using the same kinematic equation as before, we can calculate the distance:

distance = (initial velocity * time) + (0.5 * acceleration * time^2)

Again, the initial velocity is 0 since the ball is dropped. The acceleration is still -9.8 m/s^2. Plugging in the values, we get:

distance = 0.5 * (-9.8) * (2.13)^2

Solving this equation, we find that the distance from the bottom of the window to the sidewalk is approximately 22.4 meters.

Now, let's add up the distances to find the total height of the building:

H = 1.29 + 22.4

Simplifying, we find that the height of the building is approximately 23.7 meters.

To determine the height of the building, we'll need to calculate the time it takes for the ball to fall from the top of the building to the bottom of the window and then use this time to find the initial velocity of the ball.

Step 1: Calculate the time it takes for the ball to fall from the top of the building to the bottom of the window.
Given:
- Time taken for the fall from top to bottom of the window: t = 0.13 s
- Distance fallen during this time: d = 1.20 m

We can use the equation of motion for free fall: d = (1/2) * g * t^2, where g is the acceleration due to gravity (approximately 9.8 m/s^2).

Substituting the given values into the equation, we can solve for t:
1.20 m = (1/2) * (9.8 m/s^2) * (0.13 s)^2

2.13 m = (1/2) * (9.8 m/s^2) * (t + 0.13 s)^2

Simplifying further, we get:
0.065 m = (1/2) * (9.8 m/s^2) * (0.13 s)^2

Dividing both sides by (1/2) * (9.8 m/s^2), we find:
0.065 m / (0.00635 m/s^2) = (0.13 s)^2

Taking the square root of both sides gives us:
0.65 s = 0.13 s

So, the time taken for the ball to fall from the top to the bottom of the window is 0.13 s.

Step 2: Calculate the initial velocity of the ball.
The initial velocity of the ball can be found using the formula: v = g * t, where v is the initial velocity and g is the acceleration due to gravity.

Substituting the values, we have:
v = (9.8 m/s^2) * (0.13 s)
v = 1.274 m/s

So, the initial velocity of the ball is approximately 1.274 m/s.

Step 3: Calculate the height of the building.
To find the height of the building, we'll use the equation of motion for free fall: d = v0 * t + (1/2) * g * t^2, where d is the distance fallen, v0 is the initial velocity, g is the acceleration due to gravity, and t is the time of fall.

Substituting the given values, we have:
h = (1/2) * (9.8 m/s^2) * (0.13 s)^2 + (1/2) * (9.8 m/s^2) * (2.13 s)^2
h = 0.083585 m + 22.05954 m
h ≈ 22.143 m

Therefore, the height of the building is approximately 22.143 meters.

figure the average velocity in the window (distance/time). That is the velocity in the center of the window. call that Va

Va^2=2gd solve for d, the distance from the top of the building to the center of the window.

Now do the same for the upward flight.
Notice it took 2.13 sec/2 to hit the ground, so
hf=Va*1.065 -4.9*1.065^2 solve for hf, the distance below the center of the window.
How tall is the building? hf+d