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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:

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:

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

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:
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}):
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}.
"we will assume the rocket’s weight does not change during this time" - that sound in the background is Tsiolkovskiy, weeping. :-) Slightly more seriously, I would assume you're going with minimum deadweight mass so the thrust/weight really ought to be a whole lot higher just before burnout. Do you have a fudge factor for that, or is it really not a problem?

ReplyDeleteOh, Tsiolkovskiy was weeping undeed.

ReplyDeleteI'm using the full loaded weight including propellant for m_full and the full loaded weight minus propellant for m_empty. Moat calcs are done with m_full and shiuld result in slightly underestimating altitude achieved. My guess is that will be countered by weather conditions and weathercocking (I wiuld like to see cp about 1.25 to 1.5 below cg, but I don't know what the team will ultimately do). I'm most concerned with shooting this thing straight up and not upward at an angle that wastes lift on verticle movement. I also don't want to walk too far to recover it....lol.

Also, I got a list of parts and specs finally emailed to me this morning. We choose parts this evening. It's a shame there was no time to analyze these and start modeling ahead of time, but I guess that's life. I still don't have specs on the acrylic we are making our fins from or even the specific type so I can look them up...