This is really an underconstrained problem. A lot of it also belongs on physics.stackexchange, or the like, or is down to specifics of your design constraints. However, I shall summarize the basics that are relevant to rocketry and space travel here.
The ratio of the final area to the throat area is the "expansion ratio" of the nozzle, and is the main factor influencing how fast your propellant leaves. At the throat, the flow is mach 1, which is a speed that will vary by propellant composition and temperature.
From there, the nozzle expands, and the increased area causes the exhaust to accelerate due the weird way in which supersonic flows work. The final velocity is given by $v_e$ in the following:
$$
v_e = \sqrt{
\left( \frac{2 \kappa}{\kappa - 1} \right)
\left( \frac{R_u}{M} T_c \right)
\left( 1 - \left(\frac{p_e}{p_c}\right)^{(\kappa-1)/\kappa} \right)
}
$$
$R_u$ is the universal gas constant. AFAICT, dry air at $T_c=293.15\text{ K}$ (i.e. $20^\circ\text{ C}$) has an average molar mass $M \approx 28.9647\text{ g/mol}$ and a heat capacity ratio $\kappa \approx 1.400$. The chamber pressure is $p_c=100\text{ PSI}\approx 689\text{ kPa}$ (use metric please!), and you're expanding it by some ratio to get a new pressure $p_e$. If the ratio is infinity, the nozzle is "ideally expanded", and $p_e=0\text{ kPa}$.
As it happens, I wrote a handy web calculator which can be applied to this case. Expanding to vacuum (i.e., "perfectly") using the above parameters will give you an exit velocity of $v_e \approx 767.5\text{ m/s}$. In practice, you'll get lower than this because you can't make an infinitely big rocket nozzle (and if you did, it wouldn't push the ball).
Of course, exit velocity is only half the story; you also need to consider mass flow (which is calculated as the density of the air at that pressure, times the hole's area, times mach 1). If you make an teensy hole and expand the flow to almost that speed, very little mass of air will escape, and this will not be strong enough to push the ball to a high speed before the end of the pipe. Conversely, if the hole is too large, plenty of mass will flow, but the flow will not accelerate very much above mach 1 (for the above, about $343\text{ m/s}$).
One needs to work out how much the gas force from the exhaust accelerates the projectile, and the length of pipe required. Indeed, there are lots of ways to extend this analysis in various ways and to various degrees of accuracy, but they get more and more into the realm of general Physics, which I won't go into here because it's off-topic. At least now you have a way to calculate the velocity and mass flow, though.
TL;DR, though: my guess is that a reasonably sized hole will work fine, even if you don't hit upon the exactly optimal size (which would be very difficult to calculate, given all the variables, anyway). Good luck with your project, even six years later! Be sure to heed the safety warnings in the very video you linked; this sort of engineering can be dangerous.
