<h2>Context</h2>
A classmate and I need to evaluate the different physical factors that one needs to take into account to predict the trajectory of a metal ball in the game of [pétanque][1]. We thought about estimating:

1) Air resistance 
2) Rolling resistance 
3) Elastic collisions 


Our idea to tackle 1)  is to create a sort of statistical indicator $\eta = \frac{1}{n}\sum_{k=1}^n \frac{|r_k-r_k'|}{|r_k-r_k''|}$ where $n$ is the number of measurements, the $r_k$s are the measurements of the actual range, $r_k'$s are the estimated ranges without drag and $r_k''$ are the estimated ranges with drag taken into account. From there $\xi = 1-\eta$ is the percentage of improvement from taking drag into account.

This "aggregate" is used to cope with the impossibility to throw the ball by hand at fixed angles and velocities. (We do not have a launching device). 

So far, we have video footage of one of us throwing the ball at different distances and angles. We exploited the chronophotographic data in a Python notebook in which we estimated the range with and without drag and compared the results with the actual measured range.

However, it appears that $\xi$ is negative over our 29 sample trajectories, which is not what we expected from a theoretical calculation: a range about a dozen centimeters shorter when taking drag into account. 

We are currently investigating several problems with our experiment that could explain this 
including:
1) We didn't use a tripod to film, meaning the trajectories may have been misrepresented for the calculations
2) There may be too few frames (a 30FPS device was used) to use polynomial regression on the points as we did.
3) By the way, is polynomial regression the best way to fit the trajectories here ? They are not quite parabolic (presumably because of drag) but were still fit using degree 2 polynomials since increasing the degree led to `numpy.polyfit` giving a RankWarning and nonsense results. The fit is used to compute the velocity of the ball (through the derivative) and its launch angle (through trigonometry).

<h2>Questions</h2>
1.Is this protocol/design experiment sound ?

2.If so, what could be causing the error ? 


  [1]: https://en.wikipedia.org/wiki/P%C3%A9tanque