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Rocketeer Rocket Equations

Equations for model rocketeers - how to accurately predict speed and altitude for your rocket from weight, diameter, motor thrust and impulse.


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V-2 How High Will My Rocket Go?

Some Quick Comments:

The Equations

There are three basic equations to find the peak altitude for your rocket (they are also summarized here):

All the weird terms in these equations are explained in the following section on the method for using the equations.

The Method

  1. Compute Some Useful Terms
    • Find the mass M of your rocket in kilograms (kg):
      M = (weight in ounces)/16/2.2
    • Find the area A of your rocket in square meters (m^2):
      A = pi*(0.5*(diameter in inches/12)*0.3048)^2 = pi*r^2
      where r is the radius in meters
    • Note that the wind resistance force = 0.5 * rho*Cd*A * v^2, where
      rho is density of air = 1.2 kg/m^3
      Cd is the drag coefficient of your rocket which is around 0.75 for a model rocket - for a really great NAR report on Cd values click here
      v is the velocity of the rocket. You don't calculate this drag force, though, since you don't know what "v" is yet. What you do need is to lump the wind resistance factors into one coefficient k:
      k = 0.5*rho*Cd*A = 0.5*1.2*0.75*A
    • Find the impulse I and thrust T (big T) of the engine for your rocket from the engine designator, for an explanation you can reference NAR's Rocket Motor Coding, the NAR Model Rocket Safety Page, which gives the nominal impulse for each category, and the Model Rocket Motor Dimension WWW Page which gives you the actual rating for specific motors.
    • Compute the burn time t (little t) for the engine by dividing impulse I by thrust T:
      t = I / T
    • Note also - the gravitational force is equal to M*g, or the mass of the rocket times the acceleration of gravity (g). The value of g is a constant, equal to 9.8 meters/sec/sec. This force is the same as the weight of the rocket in newtons, and the term M*g shows up in the following equations a lot.

  2. Compute a Couple of My Terms (to simplify the upcoming equations). In the absence of something more colorful I call them "q" and "x"
    • q = sqrt([T - M*g] / k)
    • x = 2*k*q / M

  3. You Are Ready to Go: Calculate velocity at burnout (max velocity, v), boost phase distance yb, and coast phase distance yc (you will sum these last two for total altitude). Note that "g" is acceleration of gravity in metric units = 9.8.
    • v = q*[1-exp(-x*t)] / [1+exp(-x*t)]
    • yb = [-M / (2*k)]*ln([T - M*g - k*v^2] / [T - M*g])
    • yc = [+M / (2*k)]*ln([M*g + k*v^2] / [M*g])

    If you're not familiar with the term "exp" in these equations click RIGHT HERE.

  4. AND THE TOTAL ALTITUDE IS... yb + yc

alfred Can I See A Simple Example?

Sure. Let's use an Estes Alpha III, to keep it basic:

Now let's follow the equations above and see what we get:

  1. Compute the Useful Terms
    • Mass of the rocket M = (weight in ounces)/16/2.2 = (1.2+0.7) / 16 / 2.2 = 0.05398 kg
    • Area of the rocket: A = pi*r^2 = 3.14*(0.5*0.976/12*0.3048)^2 = 0.000483 m^2
    • Compute wind resistance factor: k = 0.5*rho*Cd*A = 0.5*1.2*0.75*0.000483 = 0.000217
    • A C6 motor has a nominal impulse of 10 N-s and thrust of 6 N. The "rating" cited above applies to the impulse, giving us an actual impulse for an Estes C6 of 10*90% = 9 N-s.
    • Compute the burn time t = I / T = 9 / 6 = 1.5 sec.
    • The gravitational force = M*9.8 = 0.05398*9.8 = 0.529 newton
  2. Compute My Terms
    • q = sqrt([T - M*g] / k) = sqrt([6 - 0.05398*9.8] / 0.000217) = 158.8
    • x = 2*k*q / M = 2*0.000217*158.8 / 0.05398 = 1.277
  3. Now the good stuff:
    • v = q*[1-exp(-x*t)] / [1+exp(-x*t)]
      = 158.8*[1-exp(-1.277*1.5)] / [1+exp(-1.277*1.5)] = 118.0 m/s
      if this number doesn't mean anything to you, multiply by 2.237 to get velocity at burn-out in mph: 118.0*2.237 = 264.0 mph! And you won't get a ticket!
    • yb = [-M / (2*k)]*ln([T - M*g - k*v^2] / [T - M*g])
      = [-0.05398 / (2*0.000217)]*ln([6 - 0.05398*9.8 - 0.000217*118.0^2] / [6 - 0.05398*9.8]) = 99.95 m
      remember this is the height reached during boost. Multiply by 3.3 to get it in feet: 99.95*3.3 = 329.8 feet.
    • yc = [+M / (2*k)]*ln([M*g + k*v^2] / [M*g])
      = [+0.05398 / (2*0.000217)]*ln([0.05398*9.8 + 0.000217*118.0^2] / [0.05398*9.8])
      = 236.8 m = 781.4 feet.
      Notice: the rocket goes more than twice as far after the burn as during the burn!

  4. AND THE TOTAL ALTITUDE IS... yb + yc = 329.8 + 781.4 = 1,111 feet

V-2 Horiz What Could Possibly Go Wrong?

Good you asked. Two things will cause a difference between your calculation and where your rocket actually goes. First of all, the equations may be accurate, but ROCKETS ARE NOT! A rocket will vary in its performance for three main reasons:

Secondly, the equations make some approximations. I've minimized these as much as possible, as you will see. The three biggest approximations are

How to Handle Rocket Mass

In the example above I hand-waved the value I use for motor mass, but in reality I'm very deliberate about the value I choose. As your rocket burns fuel, the mass of the rocket changes. I estimate the mass of the rocket during the boost phase by adding

During the coast phase I use

The empty weight of the rocket is given to you as part of its specifications. You can also get the numbers on motor mass and propellant mass from the catalogs (the Estes or Aerotech catalog has their's, for instance) or you can estimate them as follows:

For fast estimation of high power (AP) motors in the field I have a cool cheat - the fully loaded motor mass in grams is roughly the same as the motor impulse in Newton-seconds. The propellant weight is roughly half the loaded motor weight. For Blackjacks add 100 grams.

For black powder (Estes) motor casings, I use a ballpark value of 0.5 ounce. I believe that's plenty good enough.

In general, to find the weight of the propellant I use a rocketry equation that states that the impulse of the motor in Newton-seconds is equal to the mass of the propellant multiplied by the exhaust velocity, or
I = M * Vex

Then the mass of the propellant, in kilograms, is the impulse divided by the exhaust velocity, for which 800 m/s is a good number for a common black-powder motor. (If you're going high power and using composites use Vex = 2000 m/s). So to get the mass of propellant just divide the impulse by 800, which gives it to you in kg. Multiply this number by 35.27 to get the propellant weight in ounces. Of course, the easy way is to get the numbers from the catalog.

Remember since you are burning the propellant up during the boost phase, use half the calculated propellant mass as an average value.

Note: if you've done any reading on rocket physics, then just plugging in an average value for the rocket mass should seem like an awful idea. THE rocket equation is derived by calculating exactly the effect of this changing mass. We can get away with using a simplification because our model rockets only have a propellant mass of 10-40% of the payload weight. By contrast, the propellent for the space shuttle is 25 times the payload weight, and the changing mass cannot be approximated by an average value.

For typical model rockets, this approximation leads to an error of less than one percent, and allows us then to use an expression that accounts for wind resistance, a much more important factor. That in fact is the secret to the accuracy of these equations.



How Long Will It Take To Get That High?

Oh, yeah, that's an easy calculation now. The time from burn-out to apogee (high point) we will call ta. To find it just calculate:

So in our example above, where M = 0.05398, k = 0.000217 and v = 118 m/s (at burnout) then

This is the time from burnout to apogee and should correlate closely with the delay time on the ejection charge on your engine. For total time launch to apogee, add burn time t and ta. The burn time we found above was t = 1.5 seconds, so the total time in our example is
t + ta = 1.5 + 5.915 = 7.415 seconds.


albert How Did You Figure This Out?

Since I posted the derivation of these equations I've received a very nice (much prettier) version in PDF format from Dr. Christian Strutz in Germany which you can get

Here is the no-frills (original) version:

Boost Phase: Velocity at Burnout

Boost Phase: Altitude at Burnout

Coast Phase: Distance Travelled from Velocity v to Zero

Coast Phase: Time to Velocity Zero

Approximation Using Static Rocket Mass

THE rocket equation assumes a dynamic mass m(t) = m0 - (dm/dt)*t, where dm/dt is a constant. When this expression is substituted into the above Second Law equations, they become intractable and must be solved with numerical methods. I therefore use a static expression for the mass M of the rocket. I will call the velocity found using the dynamic expression "vd", and the velocity found using the static expression "vs".

Define

Then we have

Then a measure of the error induced by my method is E = 1 - vs / vd. Let's suppose the extreme case (for a model rocket) that the propellant is half the total weight of the rocket, or mp = mr = m. Then
E = 1 - vs / vd = 1 - [ (m / {m+0.5*m}) / ln({m+m} / m) ] = 1 - [1 / {ln(2) * 1.5}] = 0.04, or 4% error.

You can verify for yourself that for propellants that are somewhat smaller proportions of the rocket mass, the error is much smaller. The propellant has to exceed 67% of the total rocket mass before a 10% error is induced.


Links to Other Sites

NASA NASA Space Mathematics Web Page



apollo Space Technology Home Page

How Stuff Works by Marshall Brain

launch Rocket Motion(University of Pennsylvania)


Won't Someone Do All This Calculating For Me?

Sure. Just Push The Button - This Page Is Very Cool.

launch Mark Sullivan's Model Rocket Altitude Predictor an outstanding, working online rocket simulator that does not require Java to run.

Java My Java Rocket Simulator: a running simulator and the source code used to write it. The simulator will only run if your browser is Java Enabled but you can see the source as an example in any case.

Handy spreadsheet for altitude calculation


Check The Home Page

MMU Go to Space Exploration Home Page


Rocketry Online Rocketry Online

Kiss your friends and loved ones goodbye before clicking on this link, we may never see you again.


I just recently heard from none other than Len Fehskens, who points out that in the 1960's he derived a form of the equations on this page. Shortly thereafter Doug Malewicki independently developed the same result, and the equations were named the "Fehskens-Malewicki equations". You can find references to these equations in Topics in Advanced Model Rocketry by G. K. Mandell, G. J. Caporaso, and W. P. Bengen, published in 1973 by the MIT Press, or in Estes TR-10 Altitude Prediction Charts, or you can go to Tom Beach's page on the subject. While the Fehskens-Malewicki equations are in a slightly different form, they are mathematically equivalent to the equations shown above, derived some 30 years later. Which just goes to demonstrate the old saying, "great minds think alike". Of course, there's another old saying, "idle hands are the devil's playthings". After working with the equations, you can be the judge as to which applies.


Questions

Your questions and comments regarding this page are welcome. You can e-mail Randy Culp (Tripoli #6926) for inquiries, suggestions, new ideas or just to chat.

And hey! Check out my certification level 1 rocket!

Or... my level two certification rocket - a White Sands V2.

My next project... a Warlok, bumblebee style...

Updated 25 February 2007

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