The effect of pressure on the height of the first bounce of a volleyball I'm currently working on an experiment and the first step is to come up with a hypothesis, which is based on a physics proof. The experiment involves droping a volleyball from a fixed height and observing the height of the first jump. The proof needs to show a direct correlation between pressure and the height (or displacement) of the ball after its first jump. For example:
F = ma
So the way that force can be decreased, is by decreasing the mass or decreasing the acceleration.
I started my proof by finding the velocities and connecting them to momentum. At the moment I have the image below... I'm currently stuck on this equation:
pressure = (△P x A)/△t
I'm having trouble proving how pressure affects the time the volleyball touches the ground (in theory).
Thank you for any help you provide :)

 A: The hypothesis you need to start from isn't "based on a physics proof". That's kind of putting the cart in front of the horse.
Instead, formulate a simple, reality based hypothesis like:

I assume the ball will be more resilient at higher pressure (up to a point)

Experiments using different pressure levels (and maybe different drop heights) will then allow you to determine whether or not that hypothesis is viable. They should also allow to establish reliable, statistical correlations between ball resilience and ball internal pressure.

I'm having trouble proving how pressure affects the time the volleyball touches the ground (in theory).

It's no small wonder: the ball, on impact, undergoes a complex, almost incalculable deformation, which will determine its rebound (height)
Instead try looking at the ball as an idealised Hookean spring. This is a simple model with which you can play.
As regards the formulas in your pictures, stuff gets dragged into it for reasons that aren't clear, creating a bit of 'formula salad'.
Sure (e.g.):
$$P=\frac{F}{A}$$
is the generic definitional formula for pressure but where's the context here? How does it relate to your problem?
A: I'd like to build on the strong premises of @Gert's answer, namely,

*

*A simple theoretical equation for the bounce height is probably unavailable because of the complexity of the problem, which involves air drag, friction, mechanical hysteresis, and large deflections and crumpling of the ball skin material, among other aspects.


*This is a good time to start thinking about idealized models that are as simple as possible without sacrificing the key behavior, namely, the pressure-dependent bounce height. I think an ideal spring is too simple, as I describe below, but this model can be easily revised to provide a reasonable idealization of reality.


*Combining preliminary experiments and intuition while developing the model—before ever conducting the actual experiment with a ball inflated to some pressure—is essential.
Applying our intuition, let's consider two extremes: (1) a bounce height equaling the drop height and (2) a bounce height of zero. In the first, all potential energy is stored in the ball upon deflection and converted again into kinetic energy. There are no losses. This is the ideal spring model; regardless of the stiffness, an ideal spring bounces back to its original height. (A stiffer spring simply means less deflection while in contact with the ground.) Contained gases typically act like ideal springs whose spring constant equals the instantaneous pressure. The varying spring constant doesn't introduce too much complexity here because the focus is the bounce height, which again is independent of the spring constant. A key result is that the stored energy $U$ can be modeled as obeying the equation $dU=P\,dV$, where $P$ is the instantaneous pressure and $V$ is the volume; this equation simply says that all incoming work is stored. For small changes in pressure $P$, we can integrate to obtain $\Delta U=P\Delta V$.
In the second extreme, no useful energy is stored; picture a deflected ball that simply falls to the ground without bouncing. Here, all of the original potential energy is ultimately converted into heat (including sound) and useless folding of the skin material. This behavior is often idealized using a damper or dashpot and is exemplified by crumpling  of compliant materials. The greater the degree of crumpling, the greater the dissipation of energy that cannot be used to obtain a large bounce height.
The typical inflated volleyball lies between these two extremes. A suitable model could then be the Maxwell model, which contains an ideal spring and an ideal damper in series. As we increase the pressure, we decrease the amount of deflection and crumpling that occurs in the skin material upon bouncing, and so we reduce the contribution from the damper. I'd therefore expect a smooth transition with increasing pressure.
This framework can be extended in the following ways:

*

*The pressure and volume of the ball can be related through the adiabatic relation $PV^\gamma=\mathrm{constant}$, which describes fast compression of a gas. Here $\gamma=1.4$ is the heat capacity ratio.


*The deflection and volume of the sphere can be related.. As that answer shows, the relationship is more complex than simply $P=F/A$ because the contact area changes during bouncing.


*The stored energy $\Delta U$ above can be compared to the initial potential energy $U=mgh$, where $m$ is the mass, $g$ is gravitation acceleration, and $h$ is the height.
Even without any additional theoretical work, however, you can apply this framework to qualitatively explain an increasing bounce height with increasing pressure, asymptotically approaching the original drop height (ignoring air drag). Plotted again the logarithm of pressure in your favorite units (e.g., Pa, psi, atm, or bar), I'd broadly expect the bounce height to look like a sigmoid with increasing pressure: approximately zero for low pressures and approximately the drop height for high pressures, with a transition point where the energy stored by the ideal spring (the pressurized gas) and the energy dissipated by the ideal damper (the crumpling material) are similar.
