Last instalment, I discussed the parameters of the Rocket Design Project for this semester. This time, I’m going to get into how we make our predictions. Below is a derivation of the formulas necessary for predicting the apogee of a spherical bovine rocket undergoing air drag.

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**Warning: Math Ahead**

So ultimately, we want to know how high the rocket will go. I am going to call this “y”, “displacement” or “apogee” interchangeably. I might even call it the height. Either way, this is what we are ultimately trying to solve for.

Anyone who has taken a basic calculus or physics class probably recognizes a few truths:

and

That is to say, the change in location of an object over a
period of time is its velocity, and its change in velocity over a period of
time is its acceleration. Thankfully, we can also say this in reverse:

and

This becomes important when we consider that the flight of
any object is based on the sum of the forces acting on it. This is grounded in
Newton’s Laws of Motion. Thus we know that the sum of our forces is

where F = force, m = mass, and a =
acceleration. Thus,

Now we have a few terms here that we need
to sort out: F_{Thrust}, F_{Drag}, and F_{Weight}. Of
these, we can generally not worry about F_{Thrust} because that information
is available based on our motor. Likewise, we know that F_{Weight} =
mg, where g is the acceleration due to gravity. This leaves F_{Drag} for us to define:

where ρ = the density of air, C_{D}
= the coefficient of drag, A = the crossectional area of the rocket, and v =
the velocity. This may seem like a lot of information, but much of it really
doesn’t change with time for our purposes here. So I am going to combine a
bunch of these into a constant. I will do this a few more times to clean up
equations and put them in more easily recognizable forms, too.

Now we have a simpler looking equation:

Substituting this and the formula for F

_{Weight},
we get:

One more substitution, referring back to
the beginning gives us

and we are ready to separate the
variables and integrate.

But first, we are going to use some
algebraic manipulations to get this looking like something we can integrate
more easily:

Thus

Now it’s time to integrate both sides:

Both m and k are constants, so we pull
those out and integrate the right side:

We should now recognize the integral on
the left as tanh

^{-1} u:

By the definition of tanh

^{-1},
we can perform the following manipulations:

Raising both sides to the e power gives:

Then we multiply both sides by the
denominator:

Expanding the right side gives us:

Next, I need to isolate my v’s on one
side:

And divide both sides by (-e

^{xt}-1):

And thankfully, we know that t = impulse
divided by thrust, both of which are specifications provided by our motors. So
now we know the final velocity when the motor finishes burning its fuel, but
that doesn’t tell us either how high the rocket is at that point or how far it
continues to coast. For that information, we need more integration.

We will begin by solving for y_{Burn},
or how far the rocket travels while the motor is still providing thrust. For
simplicity’s sake, we will assume the rocket’s weight does not change during
this time (we are launching a frictionless spherical cow, aren’t we?), so m
will continue to refer to the mass of the rocket while it is full of
propellant.

Referring back to our original equation,
we know that:

So we can say

By separating the variables and integrating both sides, we get:

Now we know the rocket’s altitude when
the motor cuts out, but the rocket won’t instantly stop; it will continue to
coast. So let y

_{Coast} = the distance the rocket coasts on its
momentum before beginning its descent. That is to say, how far will the rocket
travel upward once F

_{Thrust} = 0. Note that since the rocket has
exhausted all of its fuel, we no longer use m

_{Full} – we are now use m

_{Empty}.

This should all look a bit familiar, but
not that because F

_{Thrust} = 0, that term is simply missing:

By integrating both sides, we get:

Now, we know the rocket’s apogee will be
the sum of how high it goes while under power (y

_{Burn}) and how far it
coasts before the acceleration due to gravity starts returning it to Earth (y

_{Coast}):

And there you have it! To summarize:

Where,

A = the area of the rocket’s cross-section
normal to its velocity vector.

C_{D} = the rocket’s coefficient
of drag.

F_{Thrust} = the motor’s thrust
in Newtons.

g = the acceleration due to gravity.

I = the motor’s impulse in seconds.

m_{Empty} = the mass of the
rocket after burnout.

m_{Full} = the mass of the rocket
at liftoff.

ρ = the density of air.

Of course, the astute reader will notice
that through all of that mess, there is one term still undefined- C

_{D}.
This is the coefficient of drag, and determining it generally requires
experimentation or running simulations. For our purposes, that is where

OpenRocket and

SolidWorks come in. Both are capable of determining C

_{D}.

Did you survive? I warned you there’d be
math!