How High Will My Rocket Go?
Some Quick Comments:

If you want a easy way to calculate the altitude of your rockets flight
your in the wrong area, this is all to show the calculations and hence
show all the forces that act upon a rocket during its flight. Search
for wrasp or altpred these programs will do it for you but without the
fun of knowing what's really happening.

At first the equations look rough and tough but they're not to bad
once you get can see the wood from the trees.

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 rare.

This is a problem usually solved by running computer simulations.
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.

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 burnout to peak altitude.
The Equations
There are three basic equations to find the peak altitude for your rocket.

Max velocity v, the velocity at burnout = q*[1exp(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 
v is the velocity of the rocket. You don't
calculate this dragforce, 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 Standard
Motor Codes, the NAR
Model Rocket Safety Code, which gives the nominal
impulse for each category, and the NAR
High Power Rocket Safety Code, which gives the nominal
impulse for each category, and theNAR / CAR Approved Motors List 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*[1exp(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
Can I See A Simple Example?
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: C67 (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 Ns 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 Ns.

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*[1exp(x*t)] / [1+exp(x*t)]
= 158.8*[1exp(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 burnout 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
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:

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. These
have been minimized these as much as possible in the equations, 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 highpower
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 crosscheck
using a simulation.

constant motor thrust  in reality the motor thrust varies during
the burn, but the equations use a constant average value. These have
been 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 the value for mator mass had been handwaved, but
in reality a very deliberate decision was made about the value chosen.
As your rocket burns fuel, the mass of the rocket changes. The mass of
the rocket during the boost phase has been estimated 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 the equations 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 there is a
cool cheat  the fully loaded motor mass in grams is roughly the same
as the motor impulse in Newtonseconds. The propellant weight is roughly
half the loaded motor weight. For Blackjacks add 100 grams.
For black powder (Estes) motor casings, a ballpark value of 0.5 ounce
is used. This is believed to be good enough.
In general, to find the weight of the propellant you could use a rocketry
equation that states that the impulse of the motor in Newtonseconds is
equal to the mass of the propellant multiplied by the exhaust velocity,
or
I = M * Vex
alternatively research these values.
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
blackpowder 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 or 2000, 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 1040% of the payload
weight. By contrast, the propellant 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 burnout to apogee
(the highest 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.
How Did You Figure This Out?
Here it is no frills:
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: t = (M / k)*(1 / [2*q])*ln([q+v] / [qv])

Simplifying a bit: 2*k*q*t / M = ln([q+v] / [qv])
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: y = (M / 2*k)*ln([T  M*g  k*v^2] / [T 
M*g])
Coast Phase: Distance Traveled 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. Therefore we use a static expression for the mass M
of the rocket. We 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.
This page shows us what will effect how high our model rocket will go
and some of the equations we need to use to get some approximations of
probable altitude. Accurate figures are not possible due to things that
could effect this that are not easily calculated such as finish of the
rocket, wind at different altitudes, current airpressure and mony more
things.
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