To determine the time it takes for the ball to fall back to your position after being thrown upwards from a height, we can use the kinematic equation for motion under gravity:

$y(t) = y_0 + v_0 t - \frac{1}{2} g t^2$

Where:

• $y(t)$ is the position of the ball at time $t$ (in meters).
• $y_0 = 156.12 \, \text{m}$ is the initial height of the platform.
• $v_0 = 13.326 \, \text{m/s}$ is the initial velocity (upwards).
• $g = 9.81 \, \text{m/s}^2$ is the acceleration due to gravity.
• $t$ is the time in seconds.

The ball will return to the initial height $y = y_0$, so we set $y(t) = 156.12$ and solve the following equation for $t$:

$156.12 = 156.12 + 13.326 t - \frac{1}{2} (9.81) t^2$

Simplifying:

$0 = 13.326 t - 4.905 t^2$

Factoring out $t$:

$t(13.326 - 4.905 t) = 0$

This gives two solutions:

1. $t = 0$ (which corresponds to the initial time).
2. $13.326 - 4.905 t = 0$, solving for $t$:

$t = \frac{13.326}{4.905} = 2.717 \, \text{seconds}$

This is the time it takes for the ball to reach its maximum height. Now, we need to calculate the total time it takes for the ball to fall back to the platform from the highest point.

Step 2: Calculate the maximum height

Using the kinematic equation:

$y_{\text{max}} = y_0 + v_0 t - \frac{1}{2} g t^2$

At $t = 2.717 \, \text{seconds}$, we can find the maximum height reached:

$y_{\text{max}} = 156.12 + 13.326(2.717) - \frac{1}{2}(9.81)(2.717)^2$

Once we calculate the total time, I'll combine both to give the full answer.The maximum height the ball reaches is approximately $156.118 \, \text{m}$, which is very close to the original height. This suggests the ball barely gains any additional height before starting to fall back down.

Step 3: Time to fall back to the platform

Now, the total time it takes for the ball to fall back is simply the time for it to reach its maximum height plus the time it takes to fall from that point back to the original height.

Since the ball starts from rest at its highest point and falls back the same distance, we can use the following equation to calculate the fall time:

[ y = y_0 - \frac{