Ball trajectory in the game of pétanque - unexpected drag results

Question

Context

A classmate and I (about college level, not physics majors)need to evaluate the different physical factors at play during a game of pétanque. We thought about estimating the importance of air resistance during the ballistic phase first, but we are running into what we think is a problem. After some research, we decided to model air resistance with quadratic drag (the ball moves at about $5-10 \text{m}\cdot \text{s}^{-1}$).

Since we do not have any launching device to help replicate the experiments by throwing the ball at (even roughly) the same angles and speeds, we thought about quantifying the impact of 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.

Problem

However, it appears that $\xi < 0$ (i.e. $\eta>1)$ ($\xi \approx -6\% $) 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.

Possible explanations

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 30 FPS device was used) to use polynomial regression on the points as we did.

3. By the way, is polynomial regression(with least squares or any other method) 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 the overall protocol/design experiment sound?

2. What kind of fitting method could improve the numerical results?Should we try to use the model used for with-drag range calculations as a basis and then fit parameters? Wouldn't that create a bias?

Answer 1

You should be able to easily determine the drag experimentally by performing a simple vertical drop of the ball from a known height and timing the drop. Greater drop height gives better accuracy and precision. All that is needed is a stopwatch and an accurate weight of the ball. Once you know this, then the rest of your model can be evaluated further.

Your model should work for a simple vertical drop before moving on to 2 and 3d trajectories.