# Pressure vs length vs optimal speed vs volume

First off, let me say that I have spent hours trying to figure out how to calculate what I want here, but to no avail.

I guess I should go into this problem's details...

What I have here are two cylinders. I balanced the volumes of these two cylinders before experimentation for the sake of simplicity.

The first cylinder has a piston in it that exerts a certain pressure to the gas inside the cylinder, therefore changing the volume of the gas.

This pressure builds up via a stopper that happens to be a round object at the opening of the second cylinder. This object has a certain force that has to be applied to it in order for it to move (it is held into place by a small rubber stopper). Th point of this object is to be propelled by the pressure from the piston out of the second cylinder.

Now, the whole point here is that I want this round object to leave the second cylinder at the highest speed possible which means that it needs to conserve pressure behind it.

Here's my data and work so far:

Cylinder 1:

Diameter of cylinder 1 = 15.88 mm
PI = 3.14
Length of cylinder 1 = 98.73 mm
Volume of cylinder 1 = 4922.99 mm ^ 3

Volume of cylinder 1's lip/nipple that fits into cylinder 2:
506.71 mm ^ 3

Total volume of cylinder 1 (including nipple):
5429.7mm ^ 3

Cylinder 2:

Diameter of cylinder 2 = 5.98 mm
PI = 3.14
Length of cylinder 2 = 289.16 mm
Volume of cylinder 2:
5429.7mm^3

Alright now, this is nice and balanced considering room pressure and temperature but what if I decide to exert 2.0 joules of force on the piston and compress the air? How would I calculate the optimal cylinder length for speed of the round object?

This is not homework, this is a real world example that I need to figure out really soon. I will appreciate all of your help.

Thanks!

Example is attached.

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