The vomit comet bus

Check this out (saw it on the interwebs):

It's this video inside a bus of a girl hitting the roof. My first thought was: well, she just jumped. But something was odd. She was in the air too long. Well, of course this calls for a video analysis (using Tracker Video Analysis). Actually, it is a good candidate for analysis. Primarily because the motion happens right next to the back of the bus and the bus can be considered the reference frame. The only problem is the scale. I totally guessed that the back window was 3 feet, but not sure this even matters. Here is the motion of the girl with respect to the bus:

First, notice that she did indeed hit the roof (literally). You can see this in the plot of her y position. Next, let me fit a linear function to her upward motion:

Is this constant velocity or accelerating motion? It is not many data points, but from this it looks like the linear function fits well enough. Fitting a linear function for the downward motion works pretty well also. So, what does this mean? Well, if she is moving at constant speed with respect to the bus, then that part of the bus must be accelerating at near free fall motions. This is just like the vomit comet (here is a huge post about weightlessness).

Not sure what the bus did, but this is a pretty violent motion. Looking at the video, the bus is in "free fall" for at least 0.45 seconds (or at least the back of the bus - I am sure the front is still on the ground). How high would the back of the bus go if it was in the air for 0.45 seconds? I can use the kinematic equation:

There are a couple of ways to attack this problem, but my favorite trick is to just look at the motion from the highest point back to the ground. In this case, I can call the final y = 0 meters and the initial velocity is 0 m/s. This gives an initial height of:

So, putting in my 0.45 seconds, I get a height of 1 meter. Ok, that is possible. Especially if it is on a hill or something.