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.

what was the acceleration of the 2 kg cart?

⇒ 16.0 cm/s2

what was the acceleration of the 4 kg cart?

⇒ 8.1 cm/s2

what was the acceleration of the 6 kg cart?

⇒ 5.3 cm/s2

i'm on my way to history not a mistory.