You are a detective investigating why someone was hit on the head by a falling flowerpot. One piece of evidence is a smartphone video taken in a 4th-floor apartment, which happened to capture the flowerpot as it fell past a window. In a span of 8 frames (captured at 30 frames per second), the flowerpot falls 0.84 of the height of the window. You visit the apartment and measure the window to be 1.27 m tall. What can you conclude? Under what assumptions? Give as much detail as you can.

Ok, first, suppose that a given object is dropped or thrown down.

Then the acceleration of the object is the gravitational acceleration, given by:

a(t) = -9.8m/s^2

The velocity of the object can be integrated from that, it gives:

v(t) = (-9.8m/s^2)*t + V0

where V0 is the initial speed of the object, in the case that it is dropped,  V0 is equal to 0m/s

The position equation can be found integrating again:

p(t) = (1/2)*(-9.8m/s^2)*t^2 + V0*t + p0

Where p0 is the initial position.

Now let's see our problem.

We know that in 8 frames, the flowerpot falls 0.84 of the height of the window, which is 1.27m

Then the distance that it falls is:

D = 0.84*1.27m = 1.07m

Now we also know that the camera captures at 30 fps

Then we have the relation:

30 frames = 1 second

1 = (1 second)/(30 frames)

With this, we can rewrite:

8 frames = 8 frames* (1 second)/(30 frames) = 0.267 seconds.

So now we know the distance that the pot fall, and the time in which it did fall.

Remember that the position equation for a falling object, is:

p(t) = (1/2)*(-9.8m/s^2)*t^2 + V0*t + p0

let's define p0 = 0m

Then the position equation of the pot is given by:

p(t) = (1/2)*(-9.8m/s^2)*t^2 + V0*t

The distance that the pot would travel between t = 0s and t = t' is:

distance = p(t') - p(0s)

So, knowing that between t= 0s and t  = 0.267s, the pot did travel -1.07m (the negative sign is because it traveled downwards), we can find the initial speed of the pot.

-1.07m = p(0.267s) - p(0s)

-1.07m = (1/2)*(-9.8m/s^2)*(0.267s)^2 + V0*0.267s  

[-1.07m +  (1/2)*(9.8m/s^2)*(0.267s)^2]/0.267s = V0 = -2.7 m/s

This means that the pot was thrown downwards with an initial speed of 2.7 m/s

So we can conclude that the pot did not fall down on its own, someone threw it intentionally downwards.

This is the only thing that we can conclude with the given information.