By pumping air into the bottle you are increasing the pressure inside. This pressure forces the water down which in turn forces the rubber cork out of the bottle.
This force, pushing the cork, water and air out the bottom of the bottle, is the action, and it is coupled to a reaction, the bottle flying into the air. This was what Newton was referring to in his third law where he stated that for every action there is an equal and opposite reaction.
To find out why water is best to use, you can look at conservation of momentum for the entire rocket. The whole system of water and rocket has zero momentum before launch and zero momentum after launch. The heavier the water and the faster it exits the bottle, the more downward momentum it will carry. To balance this, the lighter the bottle the faster it will go, and the more spectacular the launch.
This will be a fairly fast moving projectile, so make sure you stand back while launching it. As with all projectiles, don't use anything sharp or breakable attached to the plastic bottle.
Source: NESTA
Date: 07 February 2008
Subject: Physics
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