# How does a basketball under high pressure compare to a basketball under low pressure?

Apr 8, 2016

Mass, volume, internal pressure, and bounciness?

#### Explanation:

This actually wouldn't be the worst experiment to illustrate the properties of gases, in particular that gases are massive particles (i.e. particles that have mass not big particles). Take a flat basketball, measure its mass on a suitable scale, and then measure the mass when it has been properly inflated. I have not done the math (and at the moment I have not got the basketball), but I am willing to bet that there would be a measurable difference. If you do the experiment, you should post the results here.

Apr 8, 2016

INITIAL ASSUMPTIONS

When inflating a basketball, one pumps, let's say, air, into it. There exists a volume such that the basketball does not inflate further; let us call that ${V}_{\max}$. (Let us assume that we have already stopped inflating it before it reaches an optimal internal pressure ${P}_{i n t}^{\text{opt}}$ above which it would pop.)

As $V \to {V}_{\max}$, $P$ increases. When $V = {V}_{\max}$, it's a good assumption to say that $V$ stops increasing, while $P$ continues to increase.

So, we are examining the time interval where $\textcolor{g r e e n}{\Delta V = 0}$ and $\textcolor{g r e e n}{\Delta P > 0}$.

EXAMINING THE INCREASE IN PRESSURE

Since the pressure is increasing, once $\Delta V = 0$, the gases start pushing on the inner surfaces of the basketball outwards in the radial direction (call this ${F}_{g a s}$).

As ${F}_{g a s}$ increases, we know that since $\Delta V = 0$, the basketball cannot inflate any more, so it stiffens up more and more while the pressure inside the ball increases (the gases embrace the inner surface of the ball).

FINAL INTERPRETATION

In the end, assuming that $\Delta V = 0$ and $P$ has been increasing, we have basically this image and the following interpretation:

• Before it reaches some optimal internal pressure ${P}_{i n t}^{\text{opt}}$ for optimum bounce, ${F}_{g a s}$ is not as large as it can be, and the ball bounces less high than it can. That is because a smaller ${F}_{g a s}$ corresponds to a smaller normal force ${F}_{N}$ that is exerted upwards by the floor, and thus a smaller bounce ($0 \le {F}_{g a s} \le {F}_{g a s}^{\text{opt}}$).

• At ${P}_{i n t}^{\text{opt}}$, ${F}_{g a s} = {F}_{g a s}^{\text{opt}}$ is as large as possible (but not so large that the ball pops from too much pressure), so when you bounce the basketball, the floor can exert an optimal, maximized normal force ${F}_{N}^{\text{opt}}$ to send it back upwards effectively. Hence, it is bouncier. It also has a greater mass of gas (though approximately the same volume).