Chapter 3: METHODOLOGY

3.1 Materials:

a) Rocket 1.5 liter soda bottles with standard bottle opening Cardboard or Plastic Fins Plasticine

b) Launcher Thick polystyrene box Cardboard

c) Locking system Big nails Small drinking bottle Wire Rubber band

3.2 Apparatus:

Cloth Tape and selefon tape Scissors and knifes

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3.3 Procedure:

a) Rocket making

Two bottles were prepared for making the rocket. Take one of them and cut-off its top cone to use as the head of rocket. Put in certain amount of plasticine at the rocket head cone as weight. On the other hand, three rocket fins were prepared and pasted on the second bottle using cloth tape. After this, combine the second bottle with the first bottle’s top cone using cloth tape.

Figure 1: Water rocket with its fins.

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b) Launcher making

Thick polystyrene box was cut into the desired shape for the rocket launcher. The cardboard was used to make a triangle support at the in front of the launcher to support the weight of the rocket.

Figure 2: The launching system.

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c) Locking system making

The locking system for the rocket uses a simple plastic bottle ring with two big nails and wire to hold the rocket before it launched. Below showing the draft of the locking system used in our rocket launching system:

Figure 3: The locking system.

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d) Rocket Launching

To launch the rocket, firstly, fill the rocket with around 45-50% of water. Secondly, slot in the cork with the rubber pipe. Then, place the locking system on the cork and lock it with the two nails at the side of the bottle mouth. Using a rubber band to tie the head of the nails, at the other end of the nails, using a wire to hold the nails and make it longer to around 15meter, as the rocket releaser when the pressure reached 5atm.

The next step is placing the whole rocket with the locking system on the rocket launcher by placing the locking system on the hole made on the rocket launcher.

After everything setup has been done, pressurize the water rocket up to until 5atm using pressure pump. When the pressure reached 5atm, stop pressurizes the rocket and pulls the 15meter rocket releaser wire and the rocket shall launch.

Notice the flight of the rocket and the time of flight and distance of rocket fall shall be jotted down.

(a) (b)

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Figure 4: (a) The side view of the rocket setup before launch, (b) the top view of the rocket setup.

Chapter 4: RESULTS

Mass of rocket (with water fill in): 2.5kg

Pressure in rocket: 5atm

Time of flight: 9.5s

Distance acquired: 95meter

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Chapter 5: DISCUSSIONS.

5.1 Rocket design

Using 1.5liter cylindrical water bottles as the pressure chamber (rocket body) as the

cylindrical shape allows better air flow and lesser air resistance.

Fins are the guidance system for the rocket. Without them a rocket would tumble end

over end. With the incredible speeds and frightful acceleration generated at launch, many

fins get ripped off the rocket body within a fraction of a second. We have found that tape

(silver, duct, gaffa or cloth) works the best at holding the fins on the rocket as this allows

for the flexing of the rocket body.

To ensure stability and safety, the minimum number of fins on a rocket is three (3). Many

people choose a 3 or 4 fin design. There is no maximum number of fins one may have but

the more fins one have the more drag one will create and drag slows a rocket down.

The fins we used are designed especially for better performance. The flow created shall

be laminar flow and not turbulent as turbulent flow shall affect the stability of the rocket.

The fins we used also having a smaller size in order to reduce the contact area through

the flow and to minimize the weight of the rocket.

This fin is way too big as the size is big and so as the contact area

with the wind flows. The big size also adding the weight to the rocket

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system.

This fin will cause turbulent flow at the edge of the rocket when

it flight. This will cause the rocket flight to be unstable and

eventually the smoothness of the flight get affected.

This is the fin we choose to use because it is small in size, so

as the contact area with the wind flow and the effect to the total

weight of rocket. The design at the edge of the fin allows

laminar flow at the edge of the rocket to make sure a smoother

and calmer flight in order to achieve highest flight travel

distance.

The nose cone is made from the cut-off top part of another water bottle. One also can

make the cone by simply cutting a large circle out of the board (about a 15cm radius). Cut

a line from the outer edge of the circle to the center on the radius. Overlap the cut edges

and turn the circle while holding one edge stationary until you get the desired cone shape.

Secure the cone with staples or tape. Attach to the bottle with tape or similar adhesive.

As it stands thios cone lacks flight stability, this can be fixed to a degree by pressing a

small lump of clay/bluetack/plasticine to the inside top of the nose cone. This will add

mass to the cone and keep the rocket from flipping end over end while in flight.

The lower the weight of your water rocket, the better it will fly. Most of the work of

designing a lightweight rigid structure has been done already by the manufacturers of the

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fantastically strong PET bottles. In order to capitalize on the strength-to-weight ratio of

the bottles, we need to avoid adding too much weight as we improve the aerodynamics of

the bottle. It is also important to add the weight in the correct places so that your rocket is

aerodynamically stable. The distribution of weight along the length of the rocket is one of

the factors which determines whether it will fly like rocket, or like a bottle. What’s the

difference?

An aerodynamically stable rocket flies

with its nose first, and should have a

flight trajectory like a beautiful smooth

arc.

Figure 5: An aerodynamically stable rocket trajectory. Notice that air-resistance tends to

make the trajectory asymmetric, with the rocket falling rather more steeply than it

ascends.

An aerodynamically unstable rocket may start out with its nose first, but its flight will

quickly become unstable and it will flap and tumble in the air, and then simply fall to

Earth.

Figure 6: An aerodynamically un-stable ‘bottle’ trajectory. Several commercially sold

rocket systems have rockets that perform in this way.

In order to make your rocket fly ‘like a rocket’ rather than ‘like a bottle’, the weight

needs to be in the front half of the rocket. However depending on the design of your fins,

this may or may not be enough to ensure aerodynamically stable flight. One of the most

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important properties of your rocket is the position of its centre of mass, sometimes called

its centre of gravity.

Figure 7: The rocket.

5.2 Launcher

The launcher should be solid and stable. The materials we used to build the launcher is

just as simple as those unwanted thick polystyrene box and cardboards. We cut of it

according to the shape we want and combine them up using cloth tape. It is light, stable

and cost effectively. We straightaway use a corner of the polystyrene box as the basic

structure for the launcher. Then using the leftover to make the support of the back of the

launcher which will make the launcher set in the desired angle for this project, which is

60º from the horizontal. This will allow the launcher to be able to last longer as it is able

to withstand the launching force. As for the front of the launcher, we use the combination

of cardboard and those leftover polystyrene parts. Using the cardboard to build a triangle

at first with the suitable angle to fit in the desired angle by the project, then at the center

of the triangle we fill in with those leftover polystyrene parts to act as parts to strengthen

the front support as it have to support the whole weight of the water rocket. A rectangular

cubic was put on the launch pad in order to set the locking system in position before a

launch.

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(a) (b)

Figure8: (a) The launcher’s side view and (b) the launcher’s top view.

5.3 Locking system

The locking system we used and apply is simple yet effective. Using a small bottle and

cut-off through its center to get a ring shape. At the both sides of the ring, we cut it from

the side with the big nail’s thickness with the depth until it meet the nozzle (bottle

mouth). Two holes were punched for the purpose of putting wire on it to hole the cork or

rubber stopper at the nozzle. When we setting up the rocket, first we put on the plastic

ring on the cork or stopper at the nozzle (bottle mouth). Then we put on two big nails at

the both sides of the plastic ring that we cut earlier on. Next we have to tie the head of the

nails with a rubber band. On the other edge of the nails, we tie them up using a wire so

that the whole locking system will be completed. The wire is then extended to about

15meter as a rocket releaser. Once the pressure in the rocket reached 5atm, the wire is

then pulled out in order to allow the rubber

band to tighten up both nails and let the

other edge of the nails is free so that the

rocket can flight.

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(a) (b)

Figure 9: (a) The locking system in side view, (b) the top view of the locking system.

5.4 Rocket Stability

Building an efficient water rocket is only part of the problem in producing a

successful rocket. The rocket must also be stable in flight. A stable rocket is one

that flies in a smooth, uniform direction. An unstable rocket flies along an erratic

path, sometimes tumbling or changing direction. Unstable rockets are dangerous

because it is not possible to predict where they will go. They may even turn upside

down and suddenly head back directly to the launch pad.

Making a rocket stable requires some form of control system. Controls can be either

active or passive. The difference between these and how they work will be explained

later. It is first important to understand what makes a rocket stable or unstable.

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All matter, regardless of size, mass, or shape, has a point inside called the center of

mass (CM). The center of mass is the exact spot where all of the mass of that object is

perfectly balanced. We can easily find the center of mass of an object such as a ruler

by balancing the object on our finger. If the material used to make the ruler is of

uniform thickness and density, the center of mass should be at the halfway point

between one end of the stick and the other. If the rulers were made of wood, and a

heavy nail was driven into one of its ends, the center of mass would no longer be in

the middle. The balance point would then be nearer the end with the nail.

Knowing the center of mass is important in rocket flight because it is around this

point that an unstable rocket tumbles. As a matter of fact, any object in flight tends to

tumble. Throw a stick, and it tumbles end over end. Throw a ball, and it spins in

flight. The act of spinning or tumbling is a way of becoming stabilized in flight. A

Frisbee will go where we want it to only if we throw it with a deliberate spin. Try

throwing a Frisbee without spinning it. If we succeed, we will see that the Frisbee

flies in an erratic path and falls far short of its mark.

In flight, spinning or tumbling takes place around one or more of three axes. They are

called roll, pitch, and yaw. The point where all three of these axes intersect is the

center of mass. For rocket flight, the pitch and yaw axes are the most important

because any movement in either of these two directions can cause the rocket to go off

course. The roll axis is the least important because movement along this axis will not

affect the flight path. In fact, a rolling motion will help stabilize the rocket in the

same way a properly passed football is stabilized by rolling (spiraling) it in flight.

Although a poorly passed football may still fly to its mark even if it tumbles rather

than rolls, a rocket will not. The action-reaction energy of a football pass will be

completely expended by the thrower the moment the ball leaves the hand. With

rockets, thrust from the engine is still being produced while the rocket is in flight.

Unstable motions about the pitch and yaw axes will cause the rocket to leave the

planned course. To prevent this, a control system is needed to prevent or at least

minimize unstable motions.

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In addition to center of mass, there is another important center inside the rocket that

affects its flight. This is the center of pressure (CP). The center of pressure exists

only when air is flowing past the moving rocket. This flowing air, rubbing and

pushing against the outer surface of the rocket, can cause it to begin moving around

one of its three axes. Think for a moment of a weather vane. A weather vane is an

arrow-like stick that is mounted on a rooftop and used for telling wind direction. The

arrow is attached to a vertical rod that acts as a pivot point. The arrow is balanced so

that the center of mass is right at the pivot point. When the wind blows, the arrow

turns, and the head of the arrow points into the oncoming wind. The tail of the arrow

points in the downwind direction.

The reason that the weather vane arrow points into the wind is that the tail of the

arrow has a much larger surface area than the arrowhead. The flowing air imparts a

greater force to the tail than the head, and therefore the tail is pushed away. There is a

point on the arrow where the surface area is the same on one side as the other. This

spot is called the center of pressure. The center of pressure is not in the same place as

the center of mass. If it were, then neither end of the arrow would be favored by the

wind and the arrow would not point. The center of pressure is between the center of

mass and the tail end of the arrow. This means that the tail end has more surface area

than the head end.

It is extremely important that the center of pressure in a rocket be located toward the

tail and the center of mass be located toward the nose. If they are in the same place or

very near each other, then the rocket will be unstable in flight. The rocket will then

try to rotate about the center of mass in the pitch and yaw axes, producing a

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dangerous situation. With the center of pressure located in the right place, the rocket

will remain stable.

Control systems for rockets are intended to keep a rocket stable in flight and to steer

it. Small rockets usually require only a stabilizing control system. Large rockets, such

as the ones that launch satellites into orbit, require a system that not only stabilizes

the rocket, but also enable it to change course while in flight.

Controls on rockets can either be active or passive. Passive controls are fixed devices

that keep rockets stabilized by their very presence on the rocket's exterior. Active

controls can be moved while the rocket is in flight to stabilize and steer the craft.

Rocket Stability Determination

A rocket that flies straight through the air is said to be a stable rocket. A rocket that veers off

course or tumbles wildly is said to be an unstable rocket. The difference between the flight of

a stable and unstable rocket depends on the positioning of the center of mass and the center of

pressure. To produce the most stable flight, the center of mass and the center of pressure should

be well separated. The center of mass should be toward the rocket's nose and the center of

pressure should be toward the rocket's tail. That is because the lower end of the rocket (starting

with the center of mass and going downward) has more surface area than the upper end

(starting with the center of mass and going upward). When the rocket flies, more air pressure

exists on the lower end of the rocket than on the upper end. Air pressure will keep the lower

end down and the upper end up. If the center of mass and the center of pressure are in the same

place, neither end of the rocket will point upward. The rocket will be unstable and tumble.

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Stability Determination Instructions

1. Tie a string loop around the middle of the rocket. Tie a second string to the first so that we

can pick it up. Slide the string loop to a position where the rocket balances. We may have to

temporarily tape the nose cone in place to keep it from falling off.

2. Draw a straight line across the scale diagram of the rocket we made earlier to show where

the ruler's position is. Mark the middle of the line with a dot. This is the rocket's center of mass.

3. Lay the rocket on a piece of cardboard. Carefully trace the rocket on the cardboard and cut it

out.

4. Lay the cardboard silhouette we just cut out on the ruler and balance it.

5. Draw a straight line across the diagram of the rocket where the ruler is. Mark the middle of

this line with a dot. This is the approximate center of pressure of the rocket.

If the center of mass is well in front of the center of pressure, the rocket should be stable.

Proceed to the swing test. If the two centers are close together, add more clay to the nosecone

of the rocket. This will move the center of mass forward. Repeat steps 2 and 3 and then proceed

to the swing test.

Swing Test

1. Tape the string loop we tied around the rocket in the previous set of instructions so that it

does not slip.

2. While standing in an open place, slowly begin swinging the rocket in a circle. If the rocket

points in the direction we are swinging it, the rocket is stable. If not, add more clay to the rocket

nose cone or replace the rocket fins with larger ones. Repeat the stability determination

instructions and then repeat the swing test.

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5.5 Calculations and formulae:

Force of rocket at launch, F=PA P= 5atm x 101.325kPa= 506.625kPa

A= 1.45 x 10-4m2

The area that the force will be applied over is the area of the bottle opening. All the bottles have a standard bottle opening which is 1.45 x10-4m2.

F=2PA

=2 x 506.625 x 1.45 x 10-4

=0.147kN

Initial velocity at x-direction, (v0)x:

x=x0+(v0)xt x=distance traveled

x0=initial distance

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(v0)x=initial velocity at x-direction

t=time of flight

95=0+(v0)x(9.5)

(v0)x=10ms-1

Initial velocity, v0:

(v0)x=v0cos60º

10=v0cos60º

v0=20ms-1

Initial velocity at y-direction, (v0)y:

(v0)y=v0sin60º

=20sin60º

=17.321ms-1

Acceleration of the rocket, a:

F=ma m=mass rocket + water

147=2.5a

a=58.8ms-2

Maximum height achieved, y:

y=y0+(v0)yt-0.5gt2

=0+17.321(9.5)-0.5(9.81)(9.5)2

=279.181m

Thus the compressed air in the bottle forces the water through a nozzle (bottle neck)

which produces the thrust required to accelerate the bottle vertically upwards. We

determine the time derivative of its vertical velocity by Newton's second law of motion:

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where:

m is the instantaneous total mass of the rocket [kg]

u is the upwards velocity [m/s]

Fthrust is the thrust force (due to the expelled water) [N]

Fdrag is the drag force from the surrounding air [N]

g is the acceleration due to gravity [9.81 m/s2]

Thrust Force, Fthrust

The thrust force is proportional to the exhaust mass flow through the nozzle times the

velocity of the exhaust relative to the rocket.

where:

is the rate of mass flow of the expelled water [kg/s]

uex is the exhaust velocity of the expelled water through the nozzle [m/s]

is the density of water [1000 kg/m3]

AN is the area of the nozzle [m2]

Fthrust=1000 x 1.45 x 10-4 x 17.3212

=43.502 N

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Bernoulli's equation can be derived from the energy equation applied to the water flowing

through the nozzle. It relates the kinetic energy of the exhaust water to the compressed air

pressure applied at the water surface.

Neglecting potential energy terms, we have:

where P is absolute pressure inside the bottle and Pa is the outside (atmospheric) pressure

[Pa]

However usurface << uex and can be neglected, thus:

Combining equations (2) and (3) above we obtain:

We now continue with Page 2 of the water rocket analysis, leading to the compressed air

volume variation differential equation. Solving this equation will allow us to evaluate the

rocket performance, leading ultimately to the altitude attained by the rocket.

Adiabatic Expansion

As the water escapes, the air volume increases, causing a decrease in pressure and a corresponding decrease in thrust. We consider this process to be adiabatic (no transfer of heat during the split-second expansion process), which allows us to relate the time variation of the pressure to that of the volume.

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The adiabatic expansion process is derived from the energy equation applied to an ideal gas, and is developed in the section on Adiabatic Expansion Analysis leading to the following equation:

where:P0 is the initial absolute pressure at liftoff [Pa]V0 is the initial volume of the compressed air [m3]k is the ratio of specific heat capacities [k = 1.4 for air]P, V are the respective time varying pressure and volume of the compressed air during the thrust phase.

P=

Compressed Air Volume Variation

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The volume variation of the compressed air due to the water escaping through the nozzle is given by:

Substituting equations 3 and 5 into equation 6 and simplifying, we obtain:

Equation 7 is the differential equation for the volume variation of the compressed air as a function of time t. It cannot be solved explicitly since the volume V is deeply embedded in a nonlinear manner in the equation, hence we resort to a numerical solution.

The numerical solution of ordinary differential equations (ODEs) is an important generic problem in engineering, and you will learn various methods of solving them (such as the Runge-Kutta methods) when you study Math 344. The approach adopted by Dr. Nielsen uses an approximate numerical integration method by replacing the derivative by a first order difference method, as follows:

where:t is the elapsed time [s], thus V(t) refers to the volume at elapsed time t

is the time step incrementand P is obtained from equation 5 as:

This leads to the following solution for V(t):

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5.6 Safety

Safety rule number 1: Keep your distance when the rocket is under pressure

A pressurized bottle can be a dangerous explosive. Always keep at least 10 to 15 meters

away from any bottle filled with pressurized air, and insist that everyone else also keep

20 meters away from the bottle. If something goes wrong with the launch and the bottle

remains filled and won’t launch, empty the bottle by disconnecting the hose from the

pump before going anywhere near the bottle. At 150 psi pressure, the bottle has the

explosive power of approximately 0.6 grams of TNT.

Always wear safety goggles/glasses if one is working closer to the pressurized bottle.

Safety rule number 2: Use a suitable bottle

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Use only PET plastic bottles that originally contained carbonated drinks. Never use

bottles that didn’t, as PET water packaging is not designed to hold pressure. Never use

metal or glass bottles, as there is the danger of shattering and shrapnel from pressurizing

and from touchdown impact. Never use or re-use a bottle that appears damaged in any

way. Never use a bottle that has visible stretch marks from previous use. Never re-use a

bottle that has been filled to the stretch-failure point.

Safety rule number 4: Rockets need space

Launch only in unpopulated areas where there is at least 150 feet clearance on all sides of

launch site.

Point the rocket to an empty space or field, do not point it to peoples.

Safety rule number 5: Rockets don’t like wind

Launch the rocket on a wind free day. Wind can have an adverse effect on the flight

characteristics of the rocket. Flight may be unstable or the rocket may be blown into

undesirable locations.

Chapter 6: CONCLUSION.

We have to calculate initial velocity, v0 by first obtaining its horizontal velocity

component, (v0)x and its vertical velocity component, (v0)y. We calculated (v0)x using

rocket’s acceleration, a. On the other hand, we used gravitational acceleration, g to

calculate (v0)y. This is because the acceleration, a is acted on horizontal direction while g

is acted in vertical direction. The acceleration will not be constant throughout entire flight

by applying concept F=ma. This is because the mass would not be constant throughout

entire flight. The mass will decrease throughout the flight as water in the rocket spilled

out. This had caused the mass, m keep varying and so as the force, F. All Newton’s laws

are applicable throughout the flight of the rocket. Newton’s First law applicable at whole

rocket flight as the external forces of air friction and gravitational acceleration force had

caused the rocket not moving in constant motion and not remain in its motion. Newton’s

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Second law is applicable at the rocket flight also as F=ma. The force on the rocket is very

depending on its mass in this case because of the equation applied. Since the mass of the

rocket decreasing as it flights as the water in it being spilled out, the force of the rocket

also decreasing. Newton’s Third law is applied at the beginning of the launch as the force

of the water being spilled out from the rocket will turn to the reaction force to launch the

rocket in order to allow the rocket to flight.

APPENDICES

References

http://csep10.phys.utk.edu/astr161/lect/history/newton3laws.html

http://microgravity.grc.nasa.gov/education/rocket/termvr.html

http://resource.npl.co.uk/docs/educate_explore/water_rockets/wr_booklet_print.pdf

http://rikkiresources.wordpress.com/2007/03/21/water-rockets/

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