An analytical, or direct, solution for computing rocket altitude has an advantage over simulation - direct computation makes it easier to study the effects of changing a single parameter.
You can easily plot curves that show what happens to your rocket's flight altitude as you increase your rocket's mass or pick an engine with higher or lower thrust.
I do that here by using an Excel spreadsheet and setting up one altitude calculation per row of the spreadsheet. I copy the calculation down about 100 rows, then in order to study one parameter I change that parameter from row to row. I then plot the curves, which are shown here on this page.
These are real studies I'm doing - I'm thinking about getting a LOC Minie Magg rocket and I'm using these studies to determine what motors I would choose to fly it.
The results may be surprising at first glance, so I discuss them a bit.
What?! This Can't Be Right!
Ah, but it is.
The right-hand part of the plot is easy enough to understand (for weight greater than 0.8 kg), as the weight of the rocket goes up, the altitude goes down. The rocket is simply harder for the motor to hoist.
So why is it that, as the rocket mass drops below 0.8 kg, the altitude DROPS instead of going up? Because most of the rocket's altitude is gained, not while it's in the boost phase, but during the coast phase. If the rocket is TOO light, then it doesn't have enough weight to overcome air resistance and the drag force slows it down faster, keeping it from reaching a higher altitude.
This is similar to the case of throwing a loosely crumpled piece of paper into the wind, and then throwing a baseball. It is precisely because the baseball weighs more that it goes farther.
To prove this, we can try running our simulation with no drag. Do this by setting rho, the air density, to a very low value (like 0.0000001) or else set Cd, the rocket's drag coefficient to a similar very low value. You can't use zero because you get a division error if you try. A very low value works fine. When you do, besides getting ridiculously high altitudes, you also see that the altitude now goes up as you reduce mass, and only goes up. Without the drag factor to deal with, the rocket can no longer be too light.
So with drag in the picture, there is an optimum weight for the rocket. Guys do add weight to their rockets sometimes for precisely this reason - their rocket may be TOO light. The Minie Magg has a rocket mass of 1.14 kg, so it's on the part of the curve where I'd prefer not to add weight. Notice that if a rocket is close to the optimum weight, it's very forgiving of slight weight additions. So, for example, it may be worthwhile to add a couple of extra coats of paint to get a smoother finish, because you lose little due to the added weight and you gain a lot by reducing the drag coefficient.
Too Weird! Can this be right... ?
Yup. It's exactly right. And what it's saying is - as you increase thrust (keeping impulse the same), you DECREASE your peak altitude. I've found this to be true for any rocket under the right conditions (see Notes below). It is more true for short, fat rockets under high impulse than for long, skinny ones under low impulse.
So how come? Well, there are two effects in play here. For obvious reasons, if the thrust is below the weight of the rocket, the rocket goes nowhere. Equally obvious, as you increase the thrust above the weight of the rocket, more and more thrust is available to accelerate the rocket and a smaller fraction is being used simply to hold the rocket up against its own weight.
Something else is going on, too, though. As you increase thrust, keeping the impulse the same, you shorten the burn time. It's fairly easy to show (and intuitive) that the burnout altitude drops directly with the burnout time. So the higher the thrust the lower the burnout height.
As I said, with higher thrust, more of the thrust is applied to accelerating the rocket, so high thrust means higher burnout velocity. In the absence of drag, even a slightly higher burnout velocity will result in a much higher coast distance. This compensates for the lower burnout altitude and results in a higher overall altitude.
But WITH drag - aha - higher burnout velocity means MUCH higher drag force. This high drag force cuts away the height advantage you should get from increased burnout velocity. So with drag in play, pretty much all you get with increased thrust is lower burnout height, and altitude drops as thrust goes up.
A Couple of Notes of Interest
There is a lower limit to the thrust your rocket can use. This minimum is the thrust that gets your rocket to 20-30 mph (10-15 m/s) before it comes off the launch rod, ensuring aerodynamic stability (usually).
The formula to find this is F/m = 0.5*v2/s, where F is total force (motor thrust minus rocket weight) in newtons, m is rocket mass in kg, v is the desired velocity = 10 m/s, and s is length of launch rod. I explain this fully on my Applied Physics page.
If the altitude curve is decreasing with thrust (see Note 2 below), the peak altitude for the thrust curve usually happens below the minimum thrust you should be using for your rocket, so if you're going for altitude you will most likely want to go with the lowest thrust, highest impulse motor you can use with your rocket.
For a given rocket, as the motor impulse goes from a low value to a high value, the thrust vs. altitude curve goes from one with increasing altitude to one with decreasing altitude. I've empirically found (by playing with these simulations) that the change over point can be found as follows.
Find the maximum velocity the rocket can achieve (at the end of boost) with the given impulse (this is approximately the impulse divided by the rocket mass in kg). Next, find the peak drag force on the rocket at that velocity. For impulses where the peak drag is less than 4 times the rocket's weight, the altitude will increase with thrust. For impulses where the peak drag is greater than 4 times the rocket's weight, the altitude will decrease with increasing thrust. When the peak drag to weight is exactly 4, the curve stays flat with thrust (except, of course, very low values of thrust).
If you prefer grinding this out as a formula, then the formula to grind is 0.5*rho*Cd*A * I2 / (m3 * g). If this is greater than 4, then you have decreasing altitude with increasing thrust and so on.
So for altitude, the best motor I can find for a Minie Magg is an Aerotech H45W (320 N-s), which will loft the rocket to nearly 1600 feet. The manufacturer, LOC, says you can get to around 1600 feet, but they don't tell you how. While it will get me a good altitude, this would be a tame flight because of the low thrust (the thrust, in fact, may be too low). In reality I'd probably go for a little better show (more roar, nice column of black smoke) and pick an Aerotech H112J (280 N-s). At a lower impulse and higher thrust I would be accepting a lower altitude, about 1000 - 1200 feet.
Editor's Note: that's precisely what I did pick for my first flight on this rocket, five months later. Click here to see my certification level 1 flight results.
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Updated 24 August 2008