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A gravity-fed hopper can only support 33.5 balls per second. Calculations are shown below.
How fast can a gravity fed hopper (as distinguished from a force-fed hopper) theoretically feed paintballs into the breech of a gun? The maximum rate would be determined by how soon it would take for gravity to move a paintball from its position just above a closed bolt, and into the breach once the bolt opens.
It is assumed that the paintball starts from a sitting position above the closed bolt; i.e., the velocity is zero. No assumption is made about the diameter of the paintball.
We need two equations to solve the problem.
V = (2gh)1/2
Where g = the acceleration due to gravity, a constant 386"/sec2
h = the height the object falls; in this case the object must fall the diameter of one paintball, typically 0.689".
The 1/2 superscript means "square root".
The second equation we need is:
t = d/V
Or distance = velocity x time; this is the basic distance equation every school child is taught. It is the time t that it takes for one ball traveling at velocity V to travel the distance of one diameter d. It is in units of seconds. This is the time per ball, or seconds per ball.
Substituting the first equation for V into the second, we get:
t = d/(2gh)1/2
h, the height that the ball falls, is the same as the diameter d:
t = d/(2gd)1/2
Rearranging terms, we get:
t = 1/(2g/d)1/2
Once the ball bottoms out in the breach, you have to add the time it takes for the bolt to force the ball into the chamber, fire it, and reopen. This time is dependent on the marker. The theoretical minimum would be based on the stresses the paintball can endure before it breaks, and that is not considered here.
The time t is in units of "seconds per ball". To get "balls per second" (bps), you take the inverse:
bps = 1/t = (2g/d)1/2
Since 2g is a constant, the final result is:
bps = (772/d)1/2
Conclusion: To calculate the maximum gravity feed rate for a hopper use the equation:
bps = (772/d)1/2 where d = ball diameter
For a given ball size of 0.689", we can calculate the maximum rate to be:
bps = ( (772"/sec2) / .689" )1/2 = 33.5 balls per second
This is the theoretical maximum based on gravity. We also have to consider that the ball above the bolt has to wait while the ball in the breech is being fired. Ideally, the bolt would contact the ball just as it was completely in the breach. And there may be gasses in the breach pushing the incoming ball back out the feed tube to consider. And you have to allow time for the ball to accelerate in the barrel before opening the bolt. These factors are determined by the design of the gun. But all this is academic.