rocket design & Multiple Integrals
/The launch of a spacecraft into orbit is now a familiar event for us. In fact, it's so familiar that it is easy to overlook how remarkable this achievement is. Countless innovations and sophisticated calculations form the foundation of modern rocket design. One simple aspect of rocketry that uses calculus is projectile motion (we have already studied space curves and this is a simple application). Although the massively powerful rocket engines may attract most of the attention, the aerodynamic properties of rockets are also absolutely critical.
One property of rocket design that is very apparent, even to the untrained eye, is a rocket's slender profile in the direction of motion. It should make sense intuitively that the wider the rocket is, the more air resistance it will encounter. The total surface area of the rocket plays a part in this, but most significantly, we need to know how much of the surface area is normal to the direction of motion, since this is what produces the bulk of the air resistance. This type of surface area calculation will require new integration skills.
In chapter 15, we extend the notion of integration to double and triple integrals of functions of several variables and examine a variety of coordinate systems. In section 15.6, we use double integrals to calculate surface areas and in Chapter 14 we use double and triple integrals to compute more complicated surface integrals. These calculations can be used to produce computer simulations of the aerodynamics of rockets, other aircraft and even things as mundane as automobiles.
A tilting rocket. A model rocket.
The reduction of air resistance is not the only design issue in rocketry. To discover a different challenge, take your rocket-shaped pen or pencil and launch it into the air. You should quickly recognize that this is not a very stable projectile shape. A real rocket is even more complicated, since its engines generate a very large thrust from the bottom of the rocket and since its center of mass is changing while in flight, as the rocket burns its fuel. The calculus in this chapter can help us with a basic design principle. If a rocket starts to tilt while in flight, it will tend to rotate about its center of mass (potentially tumbling end over end). We discuss the center of mass of three-dimensional solids in sections 15.5 and 15.6. Further, in this tilted position, the side of the rocket (where there is far greater surface area than in the nose cone) produces significant air resistance. The force of the air affects the rocket as if it were focused at a single point (see the diagram shown here). This focus point, called the center of pressure is located at the same position as the center of mass of a rocket with the same shape but constant density. If the center of pressure lies below the center of mass, the air resistance will tend to push the rocket back into alignment. If the center of pressure lies above the center of mass, then, the rocket will tend to tumble end over end.
You may have noticed that model rockets usually have large lightweight fins at the bottom. One purpose of the fins is to guarantee that the center of pressure stays below the center of mass of the rocket (see G. Harry Stine's The Handbook of Model Rocketry). The calculus we develop in this chapter can be used to gain insight into rocket design and a variety of other situations that we explore throughout the chapter (and go over some of them in class).