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Model Rocketry

Miniature astronautics is the science of the construction, design and operation of vehicles for "space travel". The rockets that we are going to build and launch in this lab may not be capable of achieving orbit, or escaping the earth's pull of gravity, but they do operate under the same scientific laws that determine the characteristics of Apollo and the Space Shuttle.

Like to be here? This is an example of a Delta2 Rocket launch from Cape Canaveral.

There are three classes of rocketeers: the professionals, the amateurs and the model rocketeers. The professionals include many levels from the astronauts at the highest to the technicians and design engineers. Amateur rocketeers are those that have been labeled the "basement bombers", trying new and different ideas, sometimes resulting in injuries. The model rocketeers, like us, include millions of people who are learning about science through the application of safe and proven concepts in rocketry.

Figure 1. Students of Physics 111 class launching their rockets

In analyzing the motion of the rocket, Newton's laws must be applied. The second law of motion is the starting place:
 

Fnet = dP/dt (1)
In normal mechanics, we worked problems where mass is constant and this is essentially a simple equation. In rocketry, however, the mass is not constant. Rather, it is decreasing as the fuel is burned:
 

dP/dt = d(mv)/dt + d(m'v')/dt (2)

dP/dt = d/dt (mv) - d/dt (mv-mu)

dP/dt = mdv/dt + vdm/dt - mdv/dt - vdm/dt + mdu
where m and v are mass and speed of the rocket, m' and v' are the mass and speed of the exhaust, and u is the speed of the exhaust relative to the rocket. This is how it is done in your textbook. Combining (1) and (2) we get:
 

m dv/dt = -u dm/dt + Fnet (3)
The external forces acting on the rocket include gravity, aerodynamic drag, constraint during takeoff, and the pressure thrust due to the unbalance in the pressure of the exhaust and the ambient pressure of the atmosphere. These look like:
 

Fgrav = Fg = mg (4)

Fdrag = Fd = 1/2 rhov2 CdA (5)
where g = 9.8 N/kg (kgm/sec2), Cd is the drag coefficient, A is the cross sectional area of the rocket (this is basically the cross sectional area of the body tube plus the leading edge area of the fins) , and rho is the density of the atmosphere.
 

Fpress = Fp = (pe - p)Ae (6)

Fconst = Fc = T - mg (7)
Here, pe is the pressure of the exhaust, p the ambient pressure, Ae the area of the exit plane of the nozzle. In flight, the pressure at the exhaust is less than the ambient pressure due to the wake formed. The rocket is held on a stationary rod (for guidance during the liftoff phase of flight) and at this point, the initial velocity is essentially zero. For our model rocket we can assume that the g, Cd, rho, and p are constant. Then we can simplify and rewrite equation (3) as:
 

m dv/dt = -dm/dt u + (pe-p)Ae - mg - 1/2 rhov2CdA + Fc (8)
Whew! You might have had some time trying to follow this development. It is readily found in any standard physics text describing rocketry motion. You do have to keep your vectors straight. The force acting upwards is the the thrust. Acting downwards is mainly the weight, the aerodynamic drag and the constraint from the engine wake. For our small rocket we can ignore the engine wake drag since it is small and the engine burns for such a short time.(These are the kinds of things you need to tell your reader in discussion. Just dropping a term might be alright but "mom" would go nuts wondering what happened to it.) If we write the constraint force as in equation (7) where T is the thrust of the engine, applying the condition that while on the rod v and dv/dt are small or approximately zero, we can write an expression for the thrust produced by the rocket engine:
 

T(t) = -dm/dt  u + (pe - p)A (9)

One could surmise from this that the effective thrust gets greater in a less dense atmosphere and is greatest in a vacuum. For a large rocket traveling through great changes in atmosphere this is a significant factor. For our situation where the rocket (do you know what rocket this was?) will only go upwards a few hundred metres at most, we can treat the density as if constant. This also applies for the acceleration of gravity. If we had a high altitude rocket we'd have to account for the density change of the atmosphere and the decrease in the acceleration of gravity. Actually, since computers can do this so easily it presents no problems to the modern rocket scientist!

In this experiment, our engines are 1/2 A6-2 engines. The code is standard in the industry. The letter indicates the total impulse produced by the engine. A B-engine produces twice the impulse as an A-engine, a C-engine produces twice that of a B-engine, etc. The first letter indicates the average thrust in Newtons. In this case the average is roughly 6 Newtons. (In this case the manufacturer value for average thrust is 5.80 N.) But the thrust really is not constant. The average thrust is the total area under the curve (thrust vs. time) divided by the total time. If you recall from your elementary calculus that the averaging technique involves the integral (area under the curve) dividied by the interval over which the integral is taken (in this case the time). The last number gives the time delay in seconds between the burnout of the engine and the ejection charge. When the rocket is flown this roughly how long the rocket will coast before the parachute is deployed. A wise rocketeer will plan to use the engine that is designed so the parachute opens at the top of the trajectory. It is intuitively obvious to the most casual observer that if the rocket is still going fast when the parachute deploys some structural damage to the rocket, the parachute or both will occur.


Figure 1. Thrust vs. Time curve for 1/2 A6-2 engine.

The above curve (Figure 1) represents the thrust-time relationship as measured by Estes Industries. You will measure the thrust-time relationship using the ULI Force Sensors. It records different voltages at various times. If you record 100 voltages per second you should have data representing forces each .01 seconds. Your instructor will help you operate this device. While model rocketing is safe, your instructor must be present for this phase. It is difficult for you to believe this, but the St. Paul Fire Department has visited the lab on one occasion when this experiment was conducted so please be careful. You'll need to wear safety glasses as a precaution and be advised that the fuel will leave a smoky sulfur smell. This rarely causes distress to anyone.

The device to record the thrust versus time data is relatively simple and was constructed by students of PH111, Introductory Physics. It consists of a block of wood that holds the rocket engine and a spring which can be arranged to push against the force sensor. The voltage, as a function of time is recorded in a memory device which can then print such values on the computer screen. You can save them to a file on a floppy or dump them to Excel. Your instructor will help you calibrate the values. The voltages represent the thrust values and the timing you will set yourselves. The engine burns for approximately 0.20 seconds and then there is the two second dely before the reverse charge is ignited (to deploy the parachute.) Once calibrated, the values of thrust versus time can be known.

You will measure thrust vs. time values for the engine your instructor provides. The manufacturer estimates a 15% uncertainty in these values, so use the values determined for the class as a whole. It is unweildy to do this for every lab group. It will generally only be done once for the class. Your job then is to take the values of Thrust vs. Time and numerically integrate to determine the total impulse of the engine. Plot force (thrust) versus time. You will use Excel to do this. The area under the curve is the total impulse, defined by:
  

        tB
I = integ   T (t) dt
        0
  
 (10)
You may approximate the area under the curve in any manner you wish. Calculate the total impulse and compare it with the value reported by Estes. Calculate the percentage difference. You can call this percent error if you like. Manufacturing tolerances just cannot be closer than the 15% recorded and still make the engines affordable to kids.

The Estes Company reports that the total impulse of the 1/2 A6-2 engine is 1.25 Newton-seconds. It also reports that the average thrust is 5.8 Newtons, the total propellant mass is 1.56g, and the engine mass (including fuel) is 15.0g. Uncertainties are ±15%. Calculate the total impulse and the average thrust using the data given. Compare these results (yes, that means calculate a % error, or rather, % difference) with those from Estes.This is an important part of the laboratory activity.

Let us consider further. We can rewrite the equation of motion by combining (9) and (8):
 

a(t,v) = (1/m(t)) { T(t) - m(t)g - 1/2 rv2 CdA } (11)
We would like to be able to predict the maximum height the rocket will travel. Obviously this will depend on your individual rocket, how smoothly you painted it, how heavy it ends up, etc. You must measure the mass of your rocket. We do not have a wind tunnel available, nor the time (and perhaps not even the inclination) to measure the coefficient of drag. It is a dimensionless number, relating the relative amounts of drag for an object of a particular shape, smoothness, etc. at a given angle of attack. The angle of attack is meaningless to us here and now and it really is not needed. Normally this is determined in a wind tunnel (one is available in the lab if a student group would like to do this as an extra lab). Somehow we must estimate what the drag coefficient is. As the result of earlier testing, the drag coefficients from each part of the rocket can be determined by adding up the effects of drag from each part:

Cd= Cd(nose cone & body) + Cd(base) + Cd(fins) + Cd(interference) + Cd(launch lug) (12)

Typical values are:

Cd (nose cone and body)
 
0.205 ± .050
 
Cd (fins)
 
0.386 ± .025
 
Cd (interference)
 
0.154 ± .020
 
Cd (base)
 
0.064 ± .015
 
Cd (launch lug)
 
0.103 ± .015
 

Cd (total)

 
0.912 ± .125

Table 1: Drag Coefficients, Alpha Rocket

Note the large % uncertainty. Uncertainties always add. They never subtract or get smaller. You will have to "wing it" as far as determining your true drag. Guesstimate a value within the uncertainty, depending on whether your individual rocket is very smooth or not, etc. For example, a smoother than normal rocket will have a drag coefficient less that the .912 while a rougher than normal rocket will have asubstantially higher value of drag coefficient. It matters ultimately in how high your rocket will go.

In the powered flight phase, we can attempt now to make some determination of altitude, then apply our elementary knowledge of coasting to get the maximum altitude of flight.This is not unlike what we have done in homework assigned spreadsheet problems. Now we're applying that expertise to a real rocket!

For the sake of simplicity, let us assume that the time increases in infinitely small increments. (For practical purposes this might be of the order of 0.01 seconds.) Using Euler's Method we can write:
 

t (next step) = t (last step) + delta t (13)
Then:
 v(t + delta t) = v(t) + a(t) delta t (14)
and:
 x (t + delta t)= x(t) + v(t) delta t + 1/2 a(t) (delta t)2 (15)

 

Using this description for the equations of motion, we can hopefully calculate the acceleration, then using standard equations of motion, determine all the characteristics of flight. We should note that a small value, say 0.01 when squared gives us 0.0001. That makes this last term small and negligible compared to the other terms. So we can ignore the last term of the equation for displacement. This is just like you did with the parachute problem.

 As a function of time, the acceleration is described by equation (11). It is a function of many things, all depending upon the time. Assuming a constant density of 1.29 kg/m3, enter your values for area and coefficient of drag into the spreadsheet to determine the trajectory during the powered phase. You will have to measure the frontal area and measure the mass of the rocket, including engine wadding, etc. using a laboratory balance. After this it will follow a parabola until the parachute deploys.

The coasting phase, (the engine has burned all the fuel so mass is now constant and the thrust zero) still has aerodynamic forces as well as gravity acting. The equation of motion should look like:
 

a(t) = -g - (1/2 rv2(t) CdA (16)

 Using your values for area and C, input appropriate values into the spreadsheet to determine the maximum height of flight.

You will test this prediction with the actual flight. Fly your rocket under the guidance of your instructor. (Thank your instructor for ordering good weather for this part of the activity.) Determine the maximum height by geometric means. This part of the lab is left to you to figure out how to measure the maximum height.

Have fun. Good flight to you! May the Force be with you!

Check the appendix of Laboratory Activities for further information regarding rocket engine data. Be sure to wear safety glasses during all static firings and actual launches! Have fun!!

PROCEDURE:

This activity takes at least two sessions to accomplish. The above information fills in the details needed to properly accomplish this activity. The following lists what you will do and need to include in your report and the discussion that should accompany it.
  1. Construct a model rocket. This takes place the first day. It will give you aday to a week to let the glue dry and to paint, decorate, etc.
  2. Measure Thrust vs. Time for the rocket engine you will use.
  3. Numerically integrate this data and estimate the impulse applied during the rocket flight. Compare this value with the commercial data. Compare not only the single value of impulse, but also how the thrust acts as a function of time.
  4. Do this for average thrust as well. This value is merely (since you estimated, numerically or otherwise, the total area under the curve) the total impulse divided by the total time of burn.
  5. Measure all the information needed to estimate the trajectory. You will need the
  6. Construct a spreadsheet to calculate your trajectory. Just input appropriate measured values.Include sufficient documentation so a non-rocket scientist knows what is going on...
  7. Fly your rocket. This is done as a class activity on the second day of this activity. Your instructor will order good weather so you can do this outside as a group. It will probably be cold. Dress warmly.
  8. You must compare the predicted values of height, etc with the actual values, so your team must measure the height. It is not very convenient to use a metre stick to directly measure height. Develop a way, using triangulation, to estimate this height. What kind of uncertainties are involved? Discuss this as you compare values.
  9. Discussion is a major part of this activity. Since you are now a rocket scientist you will have to bring your data and results to the level of the reader. Be thorough but succinct.

    10. Have FUN!!

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