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, multiple stages, etc. on a rocket, but this takes too much time and money 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 images! 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. Ideally, you would take this webpage to the next step by gathering launch speed and height data as various rocket variables are adjusted.
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. Once the top of certain water bottles is cut off, the plastic sides can be rolled into another (weaker) nose cone. If you don't care about the strength of the nose cone (perhaps you are launching two-liter bottles, which are harder to acquire and store and are more likely to crush any nose cone), just paper and tape works.
Note that a two-liter bottle may or may not be the best. Although drag is proportional to the area of the front of the rocket, adding mass under the nose cone may prevent drag from changing the acceleration much from the free-fall acceleration of gravity. The momentum that can be imparted to the rocket is proportional to the volume, but a larger rocket needs more momentum to reach a speed. I haven't tested it, but I bet that a long, thin bottle would be best: has little drag, stores a lot of energy, and can easily get torque from fins.
#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 photos below).
Extra mass such as a rock or a large bouncy solid ball such as a Superball. This is to be taped underneath a nose cone. During most of the thrust when there is much water in the bottle, this extra mass is not a large fraction of the total mass, so launch speed is not greatly affected, but it later gives the rocket inertia to cut through drag, and it raises the rocket's center of mass allowing fins to work better. For a two-liter bottle, the rubber ball I used was 53 g and had a diameter of 4.5 cm.
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 may also work if you don't put the cork in very firmly, but the floor pump allows for faster launches! My fear with the weaker handheld pump is that the rocket may need to be manually diffused if enough pressure cannot be created, though the floor pump may have this potential as well. I like a pump with a pressure gauge, but, after filling up an actual bike tire, any pump can be used to get an approximate pressure reading from the muscular force needed to operate it.
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 tiny drill bit (or nail), a hole needs to be put into the center of the cork (longways). Then temporarily 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 images below. Design and construction credit of the wood stand goes to my dad, where the 4 vertical rails can be quickly removed as they are not attached other than sitting in 4 drilled holes. Building a launch stand is somewhat optional since 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).
Optional: glue gun. Using this then tape can be very sturdy, especially for larger rockets.
Optional: launch rail. If a launch rail is desired, I used a standard metal science lab stand (mine was 1 meter). See photo for Design 2. To make this work, two string loops were hot glued then taped to the rocket. The rotational inertia of the water in the rocket might be high enough to not require a rail, but, since extra mass is added to the rocket's nose cone, the rail seemed important for safety (since there is no parachute). Also, since all bottles have the same size opening, the thrust time is proportional to the bottle volume, so, assuming that the most of the mass of the rocket is in the water, thrust distance (which is approximately given by Δx = ½ a t²) is proportional to the bottle volume. A two-liter bottle has almost 3 times the volume of Arrowhead 700-mL water bottles. A basic 30-FPS phone camera revealed that the two-liter rocket only had thrust for the first 2 or so meters (regardless of any small amount of extra mass added under nose cone) when the bottles were always 1/3 full of water and the cork was pushed in hard enough to require almost 100 psi of pressure ("measured" from the feel of the bike pump just before launch). After the rocket has lost enough water and reached a certain speed, the fins start working (assuming that the center of mass is high enough and the fins are low enough).
As for how the rocket's size affects the need for a rail, one can find that the change in angle during thrust without a rail is Δθ = Δx α / a, where α is like the average angular acceleration. Using the small-extra-mass assumption like before, we eventually get Δθ being proportional to volume divided by radius, which is larger for larger rockets. However, the average angular velocity is Δθ / t, which is inversely proportional to the radius of the rocket. The Δθ has to do with aiming the rocket, and the angular velocity affects how much the rocket will tumble following the thrust, so small rockets will tumble more during thrust, and large rockets will be harder to aim. For either case, a rail would be the solution. My main reason for using a launch rail on the two-liter bottles was for safety because we put 53 grams in the nosecone.
Color! Colored pencils, markers, glitter, paint, spray paint, etc. can turn your awesome design into something worthy of display!
Design 1. Click to enlarge
Design 2. Click to enlarge
Important design considerations...
For the first design, the top of a second water bottle (with cap) is used as the nose cone to provide inertia, strength, and weight 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, (4) has a non-crushable place to hold when inserting cork, and (5) can cut through the air better. 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. Wrapping the nose cone in tape provides a more stable (and therefore higher) flight with less drag! Giving the tip the correct shape (see previous link) using cotton balls (or perhaps heavier things) and tape allows for the highest flights!
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, though this adds weight, does not help aerodynamics (adds drag), and lowers the center of mass. Fins should not be too heavy (else the rocket is too bottom heavy to get good torque) or too large (else susceptible to wind, gravity, and drag). 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 flights. 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 flight (one that does not flip around everywhere) allows all the energy to go into upward motion and prevents drag. 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. At the cost of extra drag and not being able to use a long launch rail, the spinning stabilized the rocket for a nice straight flight much like how spinning stabilizes a frisbee. You can test to see if your fins work by dropping your unfilled rocket in the upright position from a high over 10 meters.
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 a weakness of the above design as very high air pressure is not possible. However, since achieving pressure over 150 psi can cause a water bottle to explode (dangerous!), a strong-enough person such as a large adult should not view this as a weakness in the design. From my experience making dry-ice bombs (dangerous and sometimes illegal!), you would likely notice warping of the plastic before detonation.
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 (that is, 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 (videos of launches reveal this to be 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 or messing around with this free projectile simulator. As for the rocket's trajectory, check out some code I wrote that calculates trajectories and launch angles.
