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Physicist and member of No Bucks Racing Club

P.O. Box 662

Burbank, CA 91503

©Copyright 1991

This month, we begin writing a computer program to simulate the physics of racing. Such a program is quite an ambitious one. A simple racing video game, such as "Pole Position," probably took an expert programmer several months to write. A big, realistic game like "Hard Drivin" probably took three to five people more than a year to create. The point is that the topic of writing a racing simulation is one that we will have to revisit many times in these articles, assuming your patience holds out. There are many "just physics" topics still to cover too, such as springs and dampers, transients, and thermodynamics. Your author hopes you will find the computer programming topic an enjoyable sideline and is interested, as always, in your feedback.

We will use a computer programming language called Scheme. You have probably
encountered BASIC, a language that is very common on personal computers. Scheme
is like BASIC in that it is *interactive*. An interactive computer
language is the right kind to use when inventing a program as you go along.
Scheme is better than BASIC, however, because it is a good deal simpler and also
more powerful and modern. Scheme is available for most PCs at very modest cost
(MIT Press has published a book and diskette with Scheme for IBM compatibles for
about $40; I have a free version for Macintoshes). I will explain everything we
need to know about Scheme as we go along. Although I assume little or no
knowledge about computer programming on your part, we will ultimately learn some
very advanced things.

The first thing we need to do is create a *data structure* that
contains the mathematical state of the car at any time. This data structure is a
block of computer memory. As simulated time progresses, mathematical operations
performed on the data structure simulate the physics. We create a new instance
of this data structure by typing the following on the computer keyboard at the
Scheme prompt:

`(new-race-car)`

This is an example of an *expression*. The expression includes the
parentheses. When it is typed in, it is *evaluated* immediately. When we
say that Scheme is an interactive programming language, we mean that it
evaluates expressions immediately. Later on, I show how we *define* this
expression. It is by defining such expressions that we write our simulation
program.

Everything in Scheme is an expression (that's why Scheme is simple). Every expression has a value. The value of the expression above is the new data structure itself. We need to give the new data structure a name so we can refer to it in later expressions:

`(define car-161 (new-race-car))`

This expression illustrates two Scheme features. The first is that expressions
can contain sub-expressions inside them. The inside expressions are called *nested*.
Scheme figures out which expressions are nested by counting parentheses. It is
partly by nesting expressions that we build up the complexity needed to simulate
racing. The second feature is the use of the special Scheme word `define`

.
This causes the immediately following word to become a stand-in synonym for the
value just after. The technical name for such a stand-in synonym is *variable*.
Thus, the expression `car-161`

, wherever it appears after the `define`

expression, is a synonym for the data structure created by the nested expression
`(new-race-car)`

.

We will have another data structure (with the same format) for `car-240`

,
another for `car-70`

, and so on. We get to choose these names to be
almost anything we likeNote. So, we
would create all the data structures for the cars in our simulation with
expressions like the following:

`(define car-161 (new-race-car))`

(define car-240 (new-race-car))

(define car-70 (new-race-car))

The state of a race car consists of several numbers describing the physics of
the car. First, there is the car's position. Imagine a map of the course. Every
position on the map is denoted by a pair of coordinates, *x* and *y*.
For elevation changes, we add a height coordinate, *z*. The position
of the centre of gravity of a car at any time is denoted with expressions such
as the following:

`(race-car-x car-161)`

(race-car-y car-161)

(race-car-z car-161)

Each of these expressions performs *data retrieval* on the data
structure `car-161`

. The value of the first expression is the *x*
coordinate of the car, *etc.* Normally, when running the Scheme
interpreter, typing an expression simply causes its value to be printed, so we
would see the car position coordinates printed out as we typed. We could also
store these positions in another block of computer memory for further
manipulations, or we could specify various mathematical operations to be
performed on them.

The next pieces of state information are the three components of the car's
velocity. When the car is going in any direction on the course, we can ask
"how fast is it going in the *x* direction, ignoring its motion
in the *y* and *z* directions?" Similarly, we want
to know how fast it is going in the *y* direction, ignoring the *x*
and *z* directions, and so on. Decomposing an object's velocity into
separate components along the principal coordinate directions is necessary for
computation. The technique was originated by the French mathematician Descartes,
and Newton found that the motion in each direction can be analysed independently
of the motions in the other directions at right angles to the first direction.

The velocity of our race car is retrieved via the following expressions:

`(race-car-vx car-161)`

(race-car-vy car-161)

(race-car-vz car-161)

To end this month's article, we show how velocity is computed. Suppose we
retrieve the position of the car at simulated time *t _{1}*
and save it in some variables, as follows:

`(define x1 (race-car-x car-161))`

(define y1 (race-car-y car-161))

(define z1 (race-car-z car-161))

and again, at a slightly later instant of simulated time, *t _{2}*:

`(define x2 (race-car-x car-161))`

(define y2 (race-car-y car-161))

(define z2 (race-car-z car-161))

We have used `define`

to create some new variables that now have the
values of the car's positions at two times. To calculate the average velocity of
the car between the two times and store it in some more variables, we evaluate
the following expressions:

`(define vx (/ (- x2 x1) (- t2 t1)))`

(define vy (/ (- y2 y1) (- t2 t1)))

(define vz (/ (- z2 z1) (- t2 t1)))

The nesting of expressions is one level deeper than we have seen heretofore, but
these expressions can be easily analysed. Since they all have the same form, it
suffices to explain just one of them. First of all, the `define`

operation works as before, just creating the variable `vx`

and assigning
it the value of the following expression. This expression is

`(/ (- x2 x1) (- t2 t1))`

In normal mathematical notation, this expression would read and in most computer languages, it would look like this:

`(x2 - x1) / (t2 - t1)`

We can immediately see this is the velocity in the *x* direction: a
change in position divided by the corresponding change in time. The Scheme
version of this expression looks a little strange, but there is a good reason
for it: consistency. Scheme requires that all operations, including everyday
mathematical ones, appear in the first position in a parenthesised expression,
immediately after the left parenthesis. Although consistency makes mathematical
expressions look strange, the payback is simplicity: all expressions have the
same form. If Scheme had one notation for mathematical expressions and another
notation for non-mathematical expressions, like most computer languages, it
would be more complicated. Incidentally, Scheme's notation is called Polish
notation. Perhaps you have been exposed to Hewlett-Packard calculators, which
use reverse Polish, in which the operator always appears in the *last*
position. Same idea, and advantages, as Scheme, only reversed.

So, to analyse the expression completely, it is a division expression

`(/ ...)`

whose two arguments are nested subtraction expressions

`(- ...) (- ...)`

The whole expression has the form

`(/ (- ...) (- ...))`

which, when the variables are filled in, is

`(/ (- x2 x1) (- t2 t1))`

After a little practice, Scheme's style for mathematics becomes second nature and the advantages of consistent notation pay off in the long run.

Finally, we should like to store the velocity values in our data structure. We do so as follows:

`(set-race-car-vx! car-161 vx)`

(set-race-car-vy! car-161 vy)

(set-race-car-vz! car-161 vz)

The `set`

operations change the values in the data structure named `car-161`

.
The exclamation point at the end of the names of these operations doesn't do
anything special. It's just a Scheme idiom for operations that change data
structures.

Notes 1. It so happens, annoyingly, that we can't use the word `car`

.
This is a Scheme reserved word, like `define`

. Its use is explained
later (back)