While working on your physics homework in your dorm room you notice a flowerpot fall past your window because someone has carelessly knocked it off the roof. Fortunately, you computer¡¯s video camera captured the whole thing so you can measure the total time the pot was visible through your window as T. Taking out a tape measure you find your window¡¯s height is H. You¡¯ve always wanted to know how far the roof is from the top of your window (call it D) and realize that now is a great opportunity to find out without putting yourself in a dangerous or compromising position scaling the outside of your building.

1. List all known quantities you can use to calculate the distance D.
2. List all unknown quantities you will use in deriving the distance D.
3.Write the mathematical representation of the problem and solve to find the distance D from the top of the building to the top of the window. Express your answer in terms of the known quantities or a subset of those quantities only. Make sure you show all work; a correct answer without a worked-out solution will receive no credit.

1. Known quantities:

- Total time the pot was visible through the window (T)
- Height of the window (H)

2. Unknown quantities:
- Distance from the top of the building to the top of the window (D)

3. Mathematical representation and solution:

To solve for the distance D, we can use the equation for motion under constant acceleration:

H = (1/2) * g * t²

where g is the acceleration due to gravity and t is the time the pot takes to fall from the roof to the window.

Since we know the total time the pot was visible through the window as T, we can split this total time into two parts: the time it takes for the pot to fall from the roof to the top of the window (t₁), and the time it takes for the pot to fall from the top of the window to the ground (t₂):

T = t₁ + t₂

Since the pot falls under the influence of gravity, the time it takes for the pot to fall from the roof to the top of the window is the same as the time it takes for the pot to fall from the top of the window to the ground:

t₁ = t₂

Therefore, we can rewrite the equation as:

T = 2 * t₁

Simplifying further, we find:

t₁ = T/2

Now we can substitute this value of t₁ into the equation for motion under constant acceleration:

H = (1/2) * g * (T/2)²

Simplifying and rearranging the equation, we get:

D = (H * 4) / T²

So, the distance D from the top of the building to the top of the window can be expressed as (H * 4) / T².

1. Known quantities:

- Total time the pot was visible through the window (T)
- Height of the window (H)

2. Unknown quantities:
- Distance from the top of the building to the top of the window (D)

3. Mathematical representation and solution:
We can use the equation for constant acceleration in vertical motion:

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

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

Since we want to find D, which is the distance from the top of the building to the top of the window, we need to find the time the pot took to fall past the window (t).

We can rearrange the equation for H to solve for t:

t = √(2H/g)

Now, we can substitute the known values of H and T into the equation to find D:

D = T * √(2H/g)

Make sure to show all the calculations and units for a complete solution.