Rocket Equations
Equations for model rocketeers - how to accurately
predict speed and altitude for your rocket from weight, diameter,
motor thrust and impulse.
Index to This Page
Related Pages
- Rocket Equations Quick Reference:
Fast and pretty one page summary of the equations on this
page.
- Multistage Rocket Equations:
extension of the equations on this page to the problem of
finding speed and altitude for multistage rockets.
- Rocket Simulations: explanation
of basic physics and how you can program or spreadsheet
your own simulation.
Rocket Simulations
in 2D: explanation how to include launch angle in your
calculations to find the down-range distance your rocket will
go. - Applied Physics:
some other handy equations for estimating launch velocity
and proper motor size for your rocket.
- Parachute Descent
Calculations: How do you determine the right size for your
parachute? This page shows you how to find chute size and
descent velocity.
- Thrust and Mass Studies: I plotted
some curves using these equations to show how changing a
rocket's mass or changing motor thrust affects the resulting
flight altitude.
- Barrowman Equations:
determine the stability of your rocket by calculating the
center of pressure - with spreadsheet example.
Some Quick Comments:
- At first the equations look rough and
tough but they're creampuffs. They look bad because web
page writing (html) just isn't made for writing equations
(they look better here). If you
could find this page you can handle the calculations.
- These equations include the effect of
wind resistance and therefore they're accurate.
Drag effects will have a huge effect on calculated
results (factor of two to four!) but most authors view
them as "beyond the scope of the text" and
don't tell you how to include them. Formulas with wind
resistance are so rare that I had to derive them myself
(see below).
- This is a problem usually solved by running computer
simulations. With this page you can do
it yourself by hand - COOL, HUH? Plug the
equations into a spreadsheet and compare performance of
models out of the catalogues, or see how different motors
will affect the speed, altitude, or coast time of your
rocket. You can do what I do - plug the equations into
your programmable calculator and estimate altitudes and
coast times in the field, as you're prepping your
rockets.
- Everything here is converted into metric units. The
resulting velocity is computed in meters/second, and
altitude is in meters. Multiply by 3.281 to convert to
feet/second or altitude in feet, multiply meters/second
by 2.237 to get miles per hour.
- Note that there are two phases to consider for the
calculation (with or without drag): the "boost phase"
and the "coast phase".
- Boost phase is while the rocket motor is burning,
- Coast phase is the time from engine burn-out to peak
altitude.
The Equations
There are three basic equations to find the peak altitude for
your rocket (they are also summarized here):
- Max velocity v, the velocity at burnout
= q*[1-exp(-x*t)] / [1+exp(-x*t)]
- Altitude reached at the end of boost
= [-M / (2*k)]*ln([T - M*g - k*v^2] / [T - M*g])
- Additional height achieved during coast
= [+M / (2*k)]*ln([M*g + k*v^2] / [M*g])
All the weird terms in these equations are explained in the
following section on the method for using the equations.
The Method
- 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.
- 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
- 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.
- AND THE TOTAL ALTITUDE IS... yb + yc
Sure. Let's use an Estes Alpha III, to keep it basic:
- Weight: 1.2 ounces empty. I will add 0.7 ounces for the engine.
- Diameter: 0.976 inches
- Motor: C6-7 (we'll use a big one for this rocket)
Note that the
Motor Dimension WWW Page tells us that
the Estes motor has a 90% "rating" - you'll
see what that means in a moment.
Now let's follow the equations above and see what we get:
- 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
- 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
- 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!
- AND THE TOTAL ALTITUDE IS... yb + yc =
329.8 + 781.4 = 1,111 feet
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:
- rocket motor thrust can vary by 10%
either way - this can make a big difference
- the actual drag coefficient of your
rocket depends a lot on how you built it - the shape of
the fins and the nose, the launch lugs you used, the
paint job and so on.
- your rocket's stability - if it flies
funny it flies low.
Secondly, the equations
make some approximations. I've minimized these as much as
possible, as you will see. The three biggest approximations are
- constant air density - as you go up
the air gets thinner, by about 10% for every 1,000
meters. So in other words this is not much of a concern
at all up to about 3,000 feet, after that it starts to
affect the accuracy. For model rockets don't worry about
it, for high-power flights that go to five or six
thousand feet (or more), get a rough estimate by using
these equations with an average air density, and
cross-check using a simulation.
- constant motor thrust - in reality the
motor thrust varies during the burn, but we've used a
constant average value. I've compared simulations using
constant thrust and using the actual motor thrust curve
and the effect on peak altitude calculations is
negligible. The total impulse you use for the motor has a
much bigger effect than varying the thrust.
- constant motor weight - in reality the
motor mass changes because you're burning up the
propellant. How you deal with the rocket mass is
important for getting accurate results.
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
- the empty weight of the rocket,
- the weight of the empty motor casing,
and
- half the
weight of the propellant.
During the coast phase I use
- the empty weight of the rocket, and
- the weight of the empty motor casing,
- THAT'S IT, since the propellant's all
burned up at this point.
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.
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:
- qa = sqrt(M*g / k)
- qb = sqrt(g*k / M)
- ta = arctan(v / qa) / qb
So in our example above, where M = 0.05398,
k = 0.000217 and v = 118 m/s (at burnout) then
- qa = sqrt(0.05398*9.8 / 0.000217) =
49.35
- qb = sqrt(9.8*0.000217 / 0.05398) =
0.1986
- ta = arctan(118 / 49.35) / 0.1986 =
5.915 seconds
Note: you should find arctan in radians.
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.
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
- Rocket thrust = T
- Force of gravity = M*g
- Drag force on rocket =
0.5*rho*Cd*A*v^2 = k*v^2
- Net force on rocket = F = T - M*g -
k*v^2
- Newton's Second Law: F = M*a =
M*(dv/dt) = T - M*g - kv^2
- Collecting terms: dt = M*dv / (T - M*g
- k*v^2) = (M / k)*(dv / [q^2 - v^2])
where I've defined q = sqrt([T - M*g] / k)
- Integrating both sides (finite integral from 0 to v)
and rearranging:
t = (M / k)*(1 / [2*q])*ln([q+v] / [q-v])
God this makes me wish I had something better than html
for these equations.
Anyway,
- Simplifying a bit: 2*k*q*t / M =
ln([q+v] / [q-v])
Set x = 2*k*q / M and then
- Solve for v:
v = q*[1 - exp(-x*t)] / [1 + exp(-x*t)]
Boost Phase: Altitude at Burnout
- Newton's Second Law Again
F = M*a = M*(dv/dt) = M*(dv/dy)*(dy/dt) = M*v*(dv/dy) = T
- M*g - k*v^2
- Rearranging: dy = M*v*dv / (T - M*g -
k*v^2)
- Integrating both sides (finite integral from 0 to v)
and rearranging:
y = (-M / 2*k)*ln([T - M*g - k*v^2] / [T - M*g])
Coast Phase: Distance Travelled from
Velocity v to Zero
- Newton's Second Law Yet Again
F = M*a = M*v*(dv/dy) = - M*g - k*v^2
- Rearranging: dy = M*v*dv / (- M*g -
k*v^2)
- Integrating both sides: y = [+M /
(2*k)]*ln([M*g + k*v^2] / [M*g])
Coast Phase: Time to Velocity Zero
- Newton's Second Law in the Time form
F = M*a = M*(dv/dt) = - M*g - k*v^2
- Rearranging: dt = M*dv / (- M*g - k*v^2)
- Integrating both sides: t =
([M/k]/sqrt[M*g/k])*arctan(v / sqrt[M*g/k])
Simplifies (some) to: t = sqrt(M / [k*g])*arctan(v /
sqrt[M*g/k])
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
- Vx = exhaust velocity, speed of propellant leaving rocket
- mr = mass of rocket, when EMPTY
- mp = mass of propellant (total)
Then we have
- THE rocket equation (dynamic mass):
vd = Vx * ln([mr+mp] / mr)
- The "static rocket mass" equation:
vs = Vx * (mp / mr)
- The static equation equivalent to my
method of using average rocket mass is:
vs = Vx * (mp / [mr + 0.5*mp]).
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.
NASA Space Mathematics Web Page
Space Technology Home Page
How Stuff Works by Marshall Brain
Sure. Just Push The Button
The Rocket Simulator
Free web based model rocket simulator
Or try this one (requires Flash)...
University of New Mexico's Rocket Altitude Calculator
Free web based model rocket simulator
Or geek out on this simple spreadsheet.
Handy spreadsheet for altitude calculation
Check The Home Page
Go to Space Exploration Home Page
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.
Thanks for Visiting!
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 04 August 2020
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