My goals are simplicity and good design, which is a surprisingly unique combination of goals when making rockets. I was looking online for instructions to build simple water-bottle rockets, but the designs were pathetic as the rocket would just flip around randomly both on the way up and down (even in the videos that they themselves provide). I was also unhappy with the lack of important details (for example, how does "wine cork" help me when I'm ordering supplies if I do not know what type?).
There were some very cool ideas with great instructions online, but they were sometimes very complicated. Yes, a parachute that deploys at the top is cool, but my simple and good design does not require a soft landing and, unlike a parachute, won't easily get stuck in trees. I wouldn't mind one day playing around with putting accelerometers, altimeters, cameras, barometers, etc. on a rocket, but this takes too much time and electronics for now. This idea for a launch mechanism is great, but the complicated mechanism is not something that students would have at home and does not help demonstrate the ideas of good rocket design. The design I ended up with (using corks) will not achieve such heights, but it is simple! Now, if you want very simple, you can always just buy a little toy!
Most online resources for simple rockets get into the details of how to construct the rocket with step-by-step instructions. This is not my goal. Basically, just put together the materials below with the help of the image! Then, add some water to the rocket, firmly insert the cork, then launch! As for the details, figure it out as if you were on MythBusters! The goal is to think about the design rather than just mindlessly following a bunch of steps.
What you need...
2 water bottles for each rocket. The bigger the better, and the 2 bottles do not need to be the exact same type. One of the bottles (probably the smaller one) needs its cap as this bottle will be used to create the nose cone taped to the bottom of the other bottle. Two-liter bottles give more-stable (higher!) launches than smaller bottles due to their size, but these may take some time (and space) to collect if many rockets are desired.
#9 straight wine cork. I ordered mine from Amazon. Make sure that they are made from cork (that is, not synthetic). Each rocket needs 1/2 of a cork.
Tape. Packaging tape is great for both appearance and usefulness!
Sturdy material for fins. Corrugated cardboard or foam board work nicely (a box cutter will be very useful). Card stock is far too flimsy. If box cutters are too dangerous or unavailable, consider getting mat board at somewhere like Hobby Lobby (what I used in the photo below).
Box cutter and scissors. The box cutter is for cutting the nose cone from one of the water bottles (can be done in advance by an adult) and possibly also cutting the fins. The scissors are for the tape and possibly also the fins.
Hand floor bicycle pump. You will use the ball-inflator needle that should come with the pump. Any handheld ball pump will also work, but the floor pump allows for faster and higher launches! Once the rocket is built, this (and water) is the only thing required for launching.
Various tools for preparing the cork. Each cork will need to be cut in half (not longways) with a saw or whatever. Also, with a nail or tiny drill bit, a hole needs to be put into the center of the cork (longways). Then insert the ball-inflator needle since sometimes the first time inserting causes troubles.
Material for a launch stand to make sure the rocket launches upwards. Be creative with this. See what I used in the image below (design and construction credit goes to my dad). Not shown in image, a two-by-four was eventually used to improve the stand to give it four inches of clearance from the ground. Building a launch stand is somewhat optional since a pile of gravel (or even a brave person's hand) can do the trick.
Optional: safety equipment. Eye protection during launch may be desired. Also, a work glove may be useful if doing many launches (for providing hand protection when firmly inserting the cork).
Color! Colored pencils, markers, glitter, paint, spray paint, etc. can turn your awesome design into something worthy of display!
Click to enlarge
Important design considerations...
The top of a second water bottle (with cap) is used as the nose cone to provide weight and strength so that the rocket (1) points downward on the way down, (2) lands on something not crushable allowing the rocket to be highly reusable, (3) has a higher center of mass for the fins to give good torque to prevent the thing from just flipping around everywhere, and (4) has a non-crushable place to hold when inserting cork. The streamlined shape prevents drag and prevents torque above the center of mass (which is what causes random flipping). If you are interested, here is an image I found showing the best nose cone designs to reduce drag. Wikipedia and many other sources have good information on this. Not shown in above image, wrapping the nose cone in tape provides a much more stable (and therefore higher) launch with less drag! Giving the tip the correct shape (see previous link) using cotton balls and tape allows for the highest launches!
There should be at least 3 fins. Fins should be well below the center of mass so that the torque stabilizes (torque above the center of mass actually destabilizes). Fins should be low to provide good torque, but not too low else they interfere with the launch stand or get torn off by the water+air exhaust. I have seen a design that extends the rocket beneath the nozzle using the midsection of the other water bottle (from where the nose cone came from) as protection, but this adds weight, does not help aerodynamics (adds drag), and lowers the center of mass, so I do not recommend this. Fins should not be too heavy (else too bottom heavy to get good torque) or too large (else susceptible to wind, gravity, and drag). The fins shown in the image above are actually too large, especially in the areas near the middle of the rocket! Fins are of most use when near the nozzle, where small fins do the trick. A sturdy material is needed else fins flap in all directions during flight. Card stock is not nearly sturdy enough. While being sturdy and cheap, thick corrugated cardboard may add too much drag preventing high launches. I'm not sure if it exists, but perhaps a sturdy yet easy-to-cut plastic sheet would be best? Obviously, carbon fiber would be the best design if cost were not a factor! Here is an image I found showing different options for fin designs. Having sturdy vertically-straight fins of the appropriate size and location can make a huge difference because a stable launch (one that does not flip around everywhere) allows all the energy to go into upward motion. I once had a student build fins that spiraled a bit along the bottom of the rocket causing a very neat spinning rocket! This is a rocket with a similar idea. Perhaps the spinning stabilized the rocket for a nice straight launch much like how spinning stabilizes a frisbee.
For maximum height, firmly insert the cork by twisting and pushing then pounding! This allows for maximum pressure before the cork pops off (you may need to moisten the cork to first insert it). Using corks inserted by hand is the weakest aspect of the above design as very high air pressure is not possible. By the way, achieving pressure over 100 psi can cause a water bottle to explode (not dangerous), which should not be a concern with the above design.
The hole in the cork need not be very tiny because the total outward force on the needle from the internal pressure will never be large due to the small cross-sectional area of the needle. Even if the hole is not very snug, the cork will pop out long before the needle will. However, a huge hole will leak water, so don't go nuts.
Get a good air/water ratio. More air means you can pump in more air, so more energy! More water means less energy gets wasted into blowing air because the water gives rocket some mass (inertia!) to "push off of". A fast low-mass object like air takes all the energy for itself much like how a low-mass tennis ball bouncing off of a high-mass wall does not give the wall any energy. If only a tiny amount of water is used, there is a lot of stored energy, but the rocket is like the wall, and the air is the tennis ball. The wall will not move regardless of how fast the tennis ball moves. Let's make the rocket be the ball and let the water be the wall! After a bit of Googling and math, I concluded that, under typical high-pressure initial conditions, the bottle should be at least 1/3 full of water. A fun experiment is launching the rocket only using air!
A related experiment is to stand on a skateboard or on skates, then throw a tennis ball—or an empty (air-filled) gallon jug—as fast as you can forward (the fast ball is like air leaving a rocket). Then, throw a gallon of water as hard as you can. Which scenario gives you (the rocket) the most backward momentum?
To calculate the best ratio of air to water, I did a few quick physics calculations assuming an adiabatic expansion of air from a pressure p1 (mostly determined by the cork) and volume V1 to the volume V2 of the bottle into pressure p0 of the surrounding air (p0 is no more than 15 psi). To maximize final speed of the rocket using conservation of momentum and mechanical energy, I calculated that the best V1 is about 2/3 V2 (so 1/3 is water), but this V1 should be larger (less water) if p1 is not many times larger than p0. I assumed that the air's momentum and energy negligibly affect the rocket (I only calculated the work done by air until V2 and not when it sprays out after that, and I ignore the momentum and energy of the air even before V2 is reached), and I ignored the "differential" speed of water by assuming that all the water leaves at the same speed regardless of if it leaves in the beginning or end of the spray interval (probably my least accurate assumption). By doing this physics calculation, I got approximate results in just a few minutes without spending a lot of time doing hard-core rocket science! I also calculated the minimum V1 needed to expel all the water (we do not want pressure becoming p0 before all the water leaves), and I learned that this should never be a concern unless there is far too much water. In practice, it seemed that a bit more water than my approximate value was helpful likely due to how the water moves "differentially" (which makes sense since the final small bit of high-speed water to leave provides negligible lift to the rocket for the same reason that air exhaust does not provide much lift). To explore this possibility, I used conservation of energy and momentum to do another quick calculation comparing a rocket that ejects water all at the same speed with a rocket that ejects the same water (using the same stored energy as before) giving half of the water twice the speed as the other half, and I quickly learn that the latter more realistic scenario gives the rocket a much smaller final speed, so I am mostly convinced that more water than 1/3 full can help give the rocket something to "push off of" for a higher percentage of the spray time.
After this, if you are still interested in rockets, consider building a model rocket and watching October Sky! Or, consider buying and playing Kerbal Space Program.