Making a cannon with a water bottle and lighter

A cannon is essentially a bomb with a hole in it. If you place a projectile in front of the hole, it will be ejected out incredibly fast, as the combustion creates high pressure behind the bullet. 

A simple pea shooter relies on the pressure from your mouth to push a pea down a barrel. A pressure gradient diagram illustrates this through colour - the closer you are to red, the higher the pressure:

There is a high pressure behind the barrel, and as the pea moves down the barrel, pressure is relieved. The only reason the pea has a reason to move is the universes tendency to want equilibrium. 

As there is a pressure difference between your mouth and the surrounding air, the pea will do whatever it takes to maintain an equilibrium of pressure, even if that means shooting out at high speeds.

Here, the energy transfer is pressure energy (PV) to kinetic energy (J). 

Here is another pressure gradient diagram of a gun I have made, which fires marbles at close to supersonic speeds - all from a 200ml mixture of butane and air. The butane is combusted with half a lighter positioned inside an airtight bottle with a barrel out one end. 

It is filled with butane and air , closed and then ignited. A calculated pressure of 30K PSI is briefly present in the bottle, and the only way for that pressure to equalise is by shoving the projectile out of the way. A long barrel is beneficial in this scenario as it means that the projectile is accelerated for a longer time.

Using the SUVAT equation v=at, when the initial speed u is 0,we can see that the final velocity of the bullet out the barrel is dependant on the acceleration and how long the acceleration is applied. 

We can maximise the time of acceleration exposure by increasing the length of the barrel until friction starts to slow the bullet down. Using F=ma, and assuming that F stays constant (as it is always the same explosive force that acts on the bullet), the only way to increase the acceleration of the bullet is to decrease the mass. This is why I settled on a marble for a bullet, instead of lead shot or metal, as it is much lighter. 

An amalgamation of the equations leads us to a rough guide for our estimated exit velocity, not accounting for friction:

is F=ma, then a = F/m

is v=u+at then

V(exit velocity_ = u+t(F/m)

when u=0:

V(exit velocity_ = t(F/m)

Hence the gun showed above maximises t and F, and minimises m.

A video below shows the theoretical physics paying of through a practical run: