# Why is the relation between coefficient of restitution and air pressure of a basketball logarithmic?

Perhaps this is due to the spherical shape of the ball but yea, i've conducted the experiment for school and i'm trying to work through justifying this but need assistance.

edit: here's the data

• for clarification, i have pressure as the x axis and COR as the y-axis. – Ahmed Anwer Aug 25 at 5:40
• Please include your empirical data into your question. – Gert Aug 25 at 6:12
• It might be a correlation, but not causation. I guess you are asking here about the underlying physical model which connects the two. My guess would have to do with the maximum deflection the ball exhibits during a bounce. – ja72 Aug 25 at 6:41
• 0 It probably has to do with the fact that the pressure inside the ball as a function of how much air is pumped into it follows the ideal gas law which has an exponential term in it. – niels nielsen Aug 25 at 6:51
• ideal gas law doesn't have an exponential term, right? PV =nRT that's the one know of – Ahmed Anwer Aug 25 at 17:09

The dependence of the coefficient of restitution (COR), $$e$$, on the gauge pressure of a ball, $$P_G$$, is modeled and experimentally verified in Can. J. Phys. 94: 42–46 (2016) (https://www.researchgate.net/publication/281791329_The_Coefficient_of_Restitution_of_Pressurized_Balls_A_Mechanistic_Model). The dependence the authors obtain is $$\frac{1+e^2}{(1-e^2)^2}=A P_G+B$$ (see the definitions of $$A$$ and $$B$$ in the article), so it is not exponential. However, an exponential function has the correct asymptotic behavior ($$e\rightarrow 1$$ when $$P_G\rightarrow\infty$$), so it can be a good fit.