Rocket Altitude Prediction for Anyone

Rocket Altitude Prediction for Anyone

Table of Contents

Introduction

Basic Physics

Basic terminology

Units of measure and conversions

Galileo, Sir Issac, and what science does

Rockets and flight

Single engine flight profile

Motor codes and numbers – impulse, thrust, weight, etc.

Weights: rocket segments, engine, propellant

Burn time and impulse of engine

Frontal area and coefficient of drag (how to estimate)

Basic altitude prediction

Burn rate, initial acceleration, weight at burnout

Average acceleration, speed and altitude at burnout

Average deceleration to apogee, altitude at apogee, time to apogee

Aerodynamic Drag and the mess that it makes...

Spreadsheet construction

Complicating things...

Motor thrust curves

Real coefficients of drag

Air pressure and temperature variation with altitude

Clusters

Staging and air-starts

Appendices

A1. Unit conversions

A2. Standard engine classes

A3. Popular engines and specifications

A4. Drag coefficients for common rocket shapes

Glossary

Introduction: How high will it go?

How high will it go? Sooner or later, anyone who builds and flies rockets wants to know the answer to this question. Many organized rocket clubs prefer to know, before a rocket flies at an event, at least roughly how high a rocket will go before they will allow it to be launched.

Often the rocket kit manufacturer will give you a list of recommended engines for that kit and an estimate of the altitude the rocket will attain with those engines. But if you modify the kit in some way, use different engines, are flying a custom-built design, or are just flat-out paranoid and don't believe the manufacturer’s numbers, how do you know whether you will make it under the FAA waiver or not?

PC-based computer programs, such as RockSim or WinRock, are fairly easy to use and can give very accurate results if used correctly. With these software tools you can run simulations of your rockets the day before a launch meet, but this doesn’t allow you much flexibility if something changes during the meet. You can drag a laptop computer with you to the meet, of course. Not everyone can afford that, though; and RockSim costs money as well. And there are still many things RockSim cannot do (or do easily).

If rocketry is truly to be a youth-oriented hobby, shouldn’t the tools and techniques of that hobby be as inexpensive, easy to use, and most important, have as much educational value as possible? Frankly, public science education programs badly need the kind of hands-on experience in doing science that rocketry can readily supply. Long before the trip to the launch pad, the design, construction, and finishing of rockets encourages a student to develop basic planning skills, the mastery of the materials and building techniques, and an understanding of the basic physics of flying objects as it affects rocket performance and stability. And never underestimate the ego boost of seeing a rocket one has created with one’s own hands perform reliably at a well-attended launch meet. (Boom!...Agghhh!. ...Run! Run!...).

With economy and a do-it-yourself spirit in mind, let us develop some simple, pencil-paper-calculator methods for altitude prediction that can be used either at the design table or on launch day. Later we will elaborate the basic flight model to include different flight profiles, variables of environment, and more accurate estimates of drag. All these methods can be used to construct some very slick computer spreadsheets (which I will describe in due time). But first, let's learn to do it the hard way.

Basic Rocket Science

‘Rocket Science’ is a term that illustrates the dread many people feel when asked to understand the mysterious workings of our many modern technological ‘wonders’. This is hardly fair. The fact is, rocket science is not ‘rocket science’. Anyone with a calculator and the experience of a 6th grade math student can understand and predict the altitude of a simple single stage flight with fair accuracy.

The strange way scientists use and define words has a lot to do with this technophobia. Why do they use all that obnoxious jargon, anyway? Science is an essentially democratic undertaking. A scientist must not only discover something new, he must also be able to explain what he has discovered to others: this includes both fellow scientists and the public. The other scientists need to know, as precisely as possible, exactly what the discoverer did (and didn’t do) so that they can duplicate his work and hopefully come up with ways to use his discovery, avoid his mistakes, or expand on his work. This type of communication requires the invention of a very exact language, where a word only means one particular thing. This ‘terminology’ is presumably backed up by a logical system to explain why the numbers work out; in other words, math.

The public needs to know many of the same things, but explained more clearly: What did the scientist find out, exactly? Why is the discovery important? How can it be applied to improve our lives? What led the discoverer to his success? How did he do it? Where does he (and where do we) go from here? These questions are often ignored by many scientists because it doesn’t seem like as much ‘fun’ as, well, whatever it is they normally do. Even when such questions are answered, the scientist will often forget to ‘translate’ the precise terms to which he's accustomed into words that make sense to the outsider. This happens between scientific fields, within different niches of the same field, or at worst even between individual scientists.

Understanding the language of a science is much more important than understanding the math involved. What good is it to assign a number to something if you don’t know what the number means? Is it apples, or bananas, or feet-per-second, or nuclear whammions, or what? And who would care? Okay, then; let’s tackle some of the basic terminology of rocket physics, starting with the word ‘physics’ itself.

‘Physics’ is nothing more, and nothing less, than the study of objects that move. How they move, why they move, when they move (and when they don’t). Since it can be conclusively proven that literally everything moves, or at least could, physics can easily be called a ‘science of everything’. But for now we’ll narrow it down to its relevance to rocketry: how to describe the motion of a rocket in flight, through the air, from the surface of the Earth. This type of physics is often called 'ballistics'.

‘Mass’ is easily confused with the idea of ‘weight’. An object’s mass simply attaches a number to the tendency of that object to resist changes in its motion. Your ‘weight’ is nothing else than the sensation you feel because the Earth’s gravity is trying to pull your 'mass' down to its center. Since the ground is in the way, the Earth is frustrated in this regard, but the sensation of ‘weight’ remains. On Mars, this pull (or ‘gravitational attraction’) is three times less, so one’s ‘weight’ is also three times less, even though your body (hopefully) has not lost any of its actual material, or ‘mass’. As you can see, ‘mass’ is a more general way to describe the amount of material in an object, without having to worry about where that object happens to be at the time.

‘Vector’: This is a scary-sounding and peculiar term that, at first glance, seems to describe something so trivial that one might wonder what scientists do all day. A ‘vector’ is some number combined with a direction such as up, down, north, south, towards the Sun, perpendicular to the plane of the Earth’s orbit, etc. Why does this matter? Imagine, for instance, you are standing in the wind, and you are spitting. Does it matter which way you are facing, i.e. is ‘direction’ important? If the ‘vector’ of the wind is toward your face, is spitting a wise act? According to the old saying, well…

‘Position' is more 'general’ version of a common idea, which coincidentally uses the same word. The ‘more general’ part has to do with vectors, as described above. ‘Position' not only describes how far apart two objects are, but also which direction one object is from the other. For instance, the city of Portland, Oregon is (more or less) 3000 miles west of New York City. One can be even more picky, and designate some common ‘point of reference’ by which to describe the position of objects. If your house’s street address is ‘1201 SE 6th’ this might well mean that your house is twelve blocks south of the river and six blocks east of Main Street. The point where the river crosses Main Street is the ‘point of reference’. 'Height', for instance, could be defined as 'vertical position relative to a fixed point on the ground'.

‘Velocity’: speed, or how fast an object is moving. Or almost. Velocity is more generally described as the rate of change in 'position'. Since ‘position’, as defined above, is a vector, or in other words includes a direction as well as a number, ‘velocity’ must simply mean speed (relative to a fixed point) with the added notion of direction. If one car is moving at 45 miles per hour west on a one-lane road, and another car is 45 miles per hour east on the same road, what is the difference in their velocity if they happen to meet? Hmmm… Difference means subtraction, right? But… 45 minus 45 equals 0, which sure doesn’t sound like the right answer at all. Maybe differences in ‘direction’ are also important?

Physicists use words such as ‘speed’ and ‘distance’ as more general terms to talk about the magnitude (the 'number')of a velocity or position vector. In other words, if one only wants to give a general idea of how fast something is moving but doesn’t think the direction is important, they will leave out the ‘vector’ information. For example, a satellite orbiting the earth may revolve at an averagespeed of about 7600 meters per second or 17,000 miles per hour, at a distance of about 100 miles above the earth's surface. We are talking about general characteristics of the satellite’s orbit; unless we want to specify the position and velocity of the satellite at a particular instant in time, we don’t need to supply more information.

Now that we have ‘velocity’ and ‘mass’ figured out, we can tackle a trickier idea: ‘momentum’. At first, momentum seems like a rather abstract idea, but one can get a feel for it by imagining the two vehicles above crashing into each other. One can guess that if a semi truck (with a much larger mass), moving at 45 mph to the east, collides head on with a VW Bug moving 45 mph west, the truck will end up with a new air-cooled hood ornament. In other words, the larger momentum (combination of mass and speed) of the truck will cancel the smaller momentum of the VW (due to its lower mass) and continue to move in the same direction, only maybe a little slower. Conceivably, one could imagine that if the truck were going slow enough (or the Bug fast enough), their respective momenta would cancel out exactly and both vehicles would stop dead at the point of collision.

One also might image a situation where both vehicles were traveling in the same direction and one vehicle tail-ended the other; the leading vehicle would end up moving slightly faster. Momentum has been transferred from the rear vehicle to the leading one. So you can see that, even though momentum cannot be seen or pointed at easily, nonetheless we can imagine that it exists as a useful quantity.

‘Acceleration’: this is the rate of change in velocity. When you step on the gas in your car, you are accelerating, i.e. changing the rate at which your car is covering distance. Acceleration also has a ‘vector’; if you stomp on the brakes, you are also ‘accelerating’, but since your car is traveling forward, and the brakes are exerting a force backward, your speed decreases. In other words, since your ‘velocity vector’ is forward, and the ‘acceleration vector’ (braking) is in the opposite direction, the magnitude of your velocity, or your speed, decreases.

How do these changes of speed and direction come about? Can these mysterious ‘forces’ be described as numbers without worrying about the nature of the object we’re moving? Physicists call a ‘force’ anything that causes a change in the velocity of an object, i.e. accelerates that object, or modifies the effect of another accelerating force. There are many different kinds of forces, but there are only four basic ones that really affect the flight of model rockets, and each of these acts in a different way. These forces are thrust, drag, lift and gravity; since acceleration has a direction, or ‘vector’, each of them must also act in a particular direction.

'Thrust': a force created by ‘recoil’. When a rifle bullet is fired, the rifle is pushed back against one’s shoulder; this push is often called recoil. What’s really happening, though? Since the bullet is very small but moving very fast, the rifle has to jump backwards to balance the momentum the bullet is carrying away as it leaves the barrel. Similarly, when the fuel in a rocket motor is burned, the ‘ash’ is accelerated to a very high speed, and the nozzle of the rocket projects the exhaust mostly in one direction (like the barrel of a rifle). In order for the motor (and the rocket it is attached to) to balance the momentum of the exhaust, it must move in the direction opposite to the escape path of the exhaust. So, the direction of the thrust force is usually in the direction the rocket is pointing. The amount of thrust a motor produces could be calculated by adding up the masses and velocities of every particle of ‘ash’ together to get their momentum, then figuring out how much ‘recoil’ that produces on the mass of the rocket.

'Drag': another force we can blame on ‘recoil’; in this case it is caused by the particles of air bouncing off the front-facing surface area of the rocket as it flies. When a molecule of air hits the rocket, the molecule has to slow down or stop, at least momentarily. In effect the rocket is catching the particle, and must ‘recoil’ from it, or absorb some of its momentum. Obviously the ‘fatter’ the rocket, the more particles are going to hit it, and the faster the rocket moves, the harder the air is going to hit it. Other factors also affect drag, which we will get to later.

What about direction? Drag is a force, so it must have a ‘vector’. It seems reasonable that since the rocket is ‘catching’ the air particles and slowing them down, the rocket must ‘recoil’ like your hand in a mitt when you catch a baseball: your hand will 'recoil' in the same direction the ball was moving. So the direction of the drag force is in the same direction as the air is moving, or in the opposite direction that the rocket is moving.

'Lift': this is what holds up airplanes and helicopters, and it is quite a bit weirder than the other forces we have talked about so far. Lift is created by differences in the pressure of the air as it flows around an object moving through it. The wings on airplanes are purposely designed to create uneven airflow around them; uneven flow causes local differences in air pressure, which results in a ‘lift’ force perpendicular to the direction of air flow. Even a vehicle that is meant to take advantage of lift must be carefully designed. And since lift surfaces work by obstructing airflow to some extent, they also create a lot of drag.

A rocket with excessive net lift may fly just as well backwards as forwards, resulting in erratic and often alarming flights. A good rocket designer tries minimize or eliminate lift forces, so that the rocket will fly as straight as possible, with a little drag as possible. In fact, the fins and body configuration of a typical rocket are arranged so that if the rocket is not flying straight, the shape and placement of these parts tends to create a net lift force that restores the rocket to a straight course.

In our basic altitude calculations, we will assume the lift forces acting on a rocket to be negligible. The design of glider recovery systems, however, will require us to take up the subject of lift again.

'Gravity' or 'Weight': the most obvious and at the same time the most mysterious force of all. Isaac Newton discovered that every particle of matter in the universe pulls on every other particle; he called this attractive force ‘gravity’. Large globs of matter that are very tightly packed, such as the Earth, can attract other nearby globs of matter, such as your rocket, very strongly indeed.

You may now be wondering, 'With all these different forces pushing and pulling a rocket every which way, how do you decide which one wins?' Actually, this is a very easy question to answer. Even though these forces are going in different directions (all these forces are 'vectors', remember) there are simple ways to combine all of them, sort of 'adding them together', so that one can find the resultant or ‘net’ force. The ‘net’ force what is left over after all the opposing forces cancel each other out; we use this ‘net’ force, in combination with the rocket's mass, to find out how fast (and in what direction) the rocket is accelerating at a given moment.