Context

A classmate and I need to evaluate the different physical factors at play during a game of [pétanque][1]. We thought about estimating the importance of **air resistance** (during the ballistic phase) first and we are running into *what we think* is a problem.

Since we do not have any launching device to help replicate the experiments by throwing the ball at (even roughly) the same angle and speed, we thought about quantifying air resistance with a sort of "average improvement compared to vacuum calculation" metric:

$$\eta = \frac{1}{n}\sum_{k=1}^n \frac{|r_k-r_k''|}{|r_k-r_k'|}$$

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.

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).

Questions

1.Is this protocol/design experiment sound ?

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