20.1 ph Introduction to Symbolic Computing with Mathematica
Introduction
The techniques of computational science have been developing rapidly, driven by advances in computer hardware and software. In particular, the ready availability of high-performance workstations has allowed sophisticated numerical and graphical techniques to be used routinely for computational analysis in both theory and experimentation. In this assignment you are introduced to Mathematica, a computing environment developed to do pretty much any kind of mathematics on a computer. Among other things, Mathematica allows you to manipulate symbols, numbers, data, and graphics. Thus, it is a very general program; however, speed and ease of use (and yes, elegance) sometimes leave something to be desired. Such computing environments are already used quite extensively by researchers for a wide range of serious scientific calculations. More immediately, you may find Mathematica useful for a wide range of homework sets. Beware: using Mathematica does not mean that you can turn your brain to idle; you will still need to understand your problem in depth, and you will need to adapt to a different style of work from what you would do with pencil and paper. This assignment gives you a feel for the range of capabilities found in Mathematica: the best way to learn about it is to try out a few examples. Your TA will show you how to run the application on the physlab workstations. Several Mathematica manuals are available in the lab (they are not all for the the last version, 5.1, but the program has not changed much across the last few releases); you can also bring up the manual on your screen from Mathematica's Help menu.
The Assignment, Part 1 (try to work on this during your first lab session)
1. Take a quick tour of Mathematica: visit http://www.wolfram.com/products/mathematica/ tour, or (even better) run mathematica at the shell prompt, then launch the Help Browser from the Help bar menu, and select "Tour." Continue with "A Practical Introduction to Mathematica" (get it from http://documents.wolfram.com/v5/TheMathematicaBook/ APracticalIntroductionToMathematica, or select "The Mathematica Book/A Practical Introduction. . . " in the Help Browser), which presents the same material with more detail and order (you can skim the parts that seem uninteresting or trivial). 2. Explore the functionality of Mathematica on your own, inventing your own simple problems to solve. Be a little creative with your home-made problems, and try to experiment with a large range (or even expand the range) of the functions you have seen in part 1.
Where did that grapefruit come from?
Rumor (history) has it that Caltech undergrads used a kerosene cannon to lob grapefruits onto Pasadena City College, at a distance of 1000 m. Of course, Caltech students would never do something like this, at least not the students we get these days, but the physics of this venerable legend is a good problem to feed to Mathematica. Mathematically (and Mathematically), the problem is that of a projectile subject to gravity, and to the resistance of air (atmospheric drag). Leaving apart drag for the moment, the equations of motion should be quite familiar to you: dvx = 0, dt 1 dvy = -g; dt (1)
As you can imagine, the physics of atmospheric drag is very complicated. Enter approximations! We know for a fact that the drag force is null for an object at rest, and that it grows with velocity. Under certain mathematical assumptions (of smoothness, for instance), we can then write Fdrag = -B1 v - B2 v 2 - B3 v 3 - The linear term (and other odd terms) vanish because drag does not depend on the sign of the velocity. For reasonable velocities, it turns out that the quadratic term is dominant. The resulting differential equation for the velocity is P B2 v 2 dv = - . dt mv m We estimate the coefficient B2 by the following argument: to overcome atmospheric drag, the projectile must push out of the way the volume of air directly in front of it. In a time t, the mass of air moved is mair Avt, where is the density of air, and A is the projectile's frontal area. This air is given a velocity of order v, and therefore a momentum mair v over a time t. It follows that the instantaneous force exerted by the projectile on the air (and therefore, because of Newton's third law, by the air on the projectile as a drag force) is approximately Fdrag = -Av 2 . For a spherical projectile, one finds empirically that the coefficient /2 r2 works better than the nominal area 4r2 .
Box 1: A justification of the quadratic dependence of atmospheric drag. dx dy , vy = ; dt dt which (as you know well) can be solved to yield vx = x = x0 + vx0 t, 1 y = y0 + vy0 t - gt2 . 2
(2)
(3)
Of course, real projectiles don't move quite so simply, because of drag. It turns out that the appropriate expression for the drag force in the case of spherical projectiles, with reasonable velocities, is (see Box 1) 1 (4) Fdrag - r2 v 2 , 2 where is the density of air at sea level, or approximately 1.3 kg/m3 , and where r is the radius of the projectile. This drag force can then be incorporated into the equations of motion, |Fdrag | vx dvx =- , dt m v |Fdrag | vy dvy = -g - , dt m v where v =
2 2 vx + vy .
(5)
These equations are much harder to solve by pencil and paper; we will use Mathematica.
The Assignment, Part 2 (due Oct 7 2005)
1. Using Mathematica, prove the well known result that, in the absence of drag, the optimal firing angle (the angle that yields the longer range for a given initial velocity) is 45 . Then compute the velocity components necessary to reach PCC from Caltech using the optimal firing angle, still not including drag.
2
2. In Mathematica, write a routine to integrate numerically the equations for motion without drag, and verify the above result. Superimpose several plots of the trajectory, with the same initial velocity but different firing angles, to show visually that 45 is optimal. This is your first Ph20 Beautiful PlotTM : try to arrange it so that it makes your point as clearly and boldly as possible. 3. In Mathematica, assume the grapefruit has mass 0.5 kg and radius 0.05 m. Include the acceleration due to drag in the equations of motion, and now predict where the grapefruit will land if you use the initial velocity you have just found. 4. In Mathematica, try increasing the initial velocity components and changing the firing angle until you can reach the range calculated in the drag-free case. Find the optimal firing angle (an approximate solution obtained by trial and error is acceptable). Show a visual comparison of trajectories with the same range (Caltech to PCC), but different firing angles and correspondingly different initial velocities. Again, make this graph beautiful and information-dense. 5. To go beyond trial and error, implement the following hierarchy of Mathematica functions (if needed, enlist the substantial help of your TA): (a) write a function that returns the time when the grapefruit lands (y = 0) as a function of initial angle and speed; (b) using the result of item a, write another function that returns the range (x at y = 0 and t = 0), given the initi...