Be safe. Undertake this project at your own risk.
Step 1: Materials
Wood - 1x4 or 1x8 or 1xwhatever.
Nails - will need 5, I used 6d finishing.
Either 2 8d finishing nails or something else long and thin.
String - I used kite string.
Cork - from wine bottle or something, could use a rubber bung if you have one.
Pump - preferably with pressure gauge.
Inflating needle -
Bottle - 2L soda bottle works well and can handle a lot of pressure (definitely 70 psi).
Tent stakes (optional)
Step 2: Putting Together the Base
The base is made up of a wooden platform with two upright pieces. Uprights should be at least 6" long. Stand them up lengthwise and parallel to each other and nail them in position. For extra stability, drill holes in the four corners of the platform to put tent pegs in when launching.
Step 3: The Release Mechanism
This is the hard part.
Take your cork and jam an inflating needle through it, lengthwise.
Then, take the platform and put a nail through one of the uprights, but not all the way so that it goes into the one across from it. Take the cork/needle and shove it over the protruding nail, pointing the needle up.
The next thing to do is drill holes in the uprights. These holes will hold nails/bent wire/whatever you can get your hands on that is long and thin and made of metal, that prevent the rocket from taking off before you want it to.
Using your bottle for reference, carefully measure where you will drill 4 holes in the uprights as shown. Then drill them.
Fit your bottle over the cork and put the metal pieces through the holes and see if it looks right.
Next, attach string to the metal pieces so that when you pull the string, they come out of the holes. This is how you will launch the rocket.
Step 4: Launch
Take launcher outside, preferably away from houses.
Attach launcher to ground using tent stakes.
Attach bike pump to inflating needle/cork.
Fill a bottle about a third of the way with water, maybe a little more to account for spillage.
Flip bottle over and quickly put it over the cork.
Put metal pieces into position to restrain the bottle during pumping.
PUT ON YOUR EYE PROTECTION.
Pump air into the bottle. 30 psi will give you a decent launch, though soda bottles can take a lot more than that.
Pull string attached to metal pieces to launch.
DO NOT POINT AT PEOPLE.