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26-09-88/GRAVTXT

This website contains an archive of files for the Acorn Electron, BBC Micro, Acorn Archimedes, Commodore 16 and Commodore 64 computers, which Dominic Ford has rescued from his private collection of floppy disks and cassettes.

Some of these files were originally commercial releases in the 1980s and 1990s, but they are now widely available online. I assume that copyright over them is no longer being asserted. If you own the copyright and would like files to be removed, please contact me.

Tape/disk: Home » CEEFAX disks » telesoftware9.adl
Filename: 26-09-88/GRAVTXT
Read OK: ✔
File size: 162B bytes
Load address: 0000
Exec address: FFFFFFFF
Duplicates

There is 1 duplicate copy of this file in the archive:

File contents
ANIMATION TECHNIQUES WITH GRAVITY


When writing a game, or any program in which movement of graphics sprites
is incorporated, it is often desirable to make  certain characters move
with a realism that we experience in life. Gravity is one of these
realities, and when incorporated in a program the effects can be quite
interesting (or startling when they  go wrong!).

Gravity is a force which acts on all bodies that have mass, but because a
sprite has no mass (it's being held in computer memory) we have to
introduce an artificial gravity into the animation. The introduction of
gravity has many applications which include the movement of a bouncing
ball, the slowing down and then speeding up of a sprite when it jumps up
and then down in a game, or the realistic swing of a rope in games like
HUNCHBACK. When applied in a horizontal plane the methods descibed can
also be used to simulate, for example, snooker balls on a table that has
friction.

PROJECTILE MOVEMENT

Program 1 is a general purpose ball throwing program (!). Various
parameters can be changed so that there effects upon the balls movement
can be seen by altering 9 different values. These parameters are altered
by entering the corresponding number and then, if a number is required the
user is prompted to enter a value, otherwise if the option is of a YES/NO
type, the value will toggle. The default values are such that the ball
will be thrown up in the 'air' and then return to ground. The after image
of the ball can be turned off using option 8, as can the line references.
A few points may be noted :

1. The X velocity and the Y velocity are TOTALLY independant of each
other.

2. The X velocity remains constant throughout the motion because there is
no acceleration (gravity) acting in the X plane.

3. The initial Y velocity will, in general, be large to start with, reduce
as the ball gets higher until it stops, and then it will begin to gain
speed as it falls back down to the ground.

4. By altering the X velocity it may be seen that the range of the ball is
increased because the ball can travel further to the right before it
returns to the ground.

The best way to understand what is going on is to simply play with the
program.

HOW GRAVITY IS SIMULATED

The movement of the ball uses simple animation techniques, and the
introduction of gravity is achieved on line 370. The variable 'yvel%'
holds a value that represents the Y velocity of the ball i.e. it is the
number of pixils which are jumped during subsequent plotting of the ball.
Thus initially 'yvel%' is large so that the ball appears to move fast, but
this value is slowly decreased by subtracting a value held in 'gravity' so
that the ball appears to slow down as it gets higher. Eventually 'yvel%'
will become zero, afterwhich it will become negative (i.e. a -ve
velocity), so the ball will slowly begin to move in the opposite direction
and then gather speed. The variable 'gravity' represents the rate of
change of the Y velocity (its acceleration, or 'the gravitational field
strength').

CIRCULAR MOTION AND GRAVITY - the pendulum

When bodies move under the influence of gravity and are also effected by
an external force it becomes more difficult to simulate their motion on a
computer, but the secret is to take each force in turn and program the
effects that these forces will have on the body one at a time. For
example, a true pendulum bob moves in a circlular arc and is effected by
gravity, resulting in oscillations of the pendulum. This can be simulated
on a computer by firstly making a ball move in a circle on the screen and
then, as seen in program 1, it is simply a matter of altering the velocity
of the bob at different stages of its motion.

Program 2 simulates true pendulum motion. It was written by firstly
obtaining a circle function (using two SINE waves 90 degrees out of phase
- don't worry about this), and then making a graphic character move in
this circle. Having achieved this, the initial position of the bob is
decided and then the pendulum is allowed to rotate in a circular path
initially moving downwards. The procedure 'xy' calculates the x-y position
of the bob at any angle (held in 'pos') of the circle. Thus, to obtain
circuler motion it is simply a matter of giving 'pos' consecutive values
from 0 to 360 dgrees and plotting each x-y corrdinate for the
corresponding value of 'pos'.

However, we want the pendulum to start from a horizontal position, so the
initial value of 'pos' is set to 90 degrees (line 240). The speed of the
bob is determined by the step rate of the angle through which the bob
turns (i.e. the rate of change of 'pos'). This step rate is held in the
variable 'vel' so if, for example, 'vel' = 10, then the bob would move in
a circle in jumps of 10 degrees. It is now a simple matter to introduce
gravity (as we did in program 1 ) by increasing the velocity of the bob as
it falls. This is achieved on line 360.

There is, however, one problem. Because the velocity is constantly
increasing from a positive value as it swings downwards, it will never
reach a turning point at which the velocity will begin to get slower, so
the result will be a constantly accelerating bob moving in a circle. Thus
we must put in a condition whereby once the bob reaches a certain
velocity, the sign of the gravity is changed resulting in the bob slowing
down until it eventually stops at the top of its swing (see lines
340,350). Alternatively, the sign of 'gravity' could be swapped when the
value of 'pos' equaled 180 degrees, but you have to make sure that 'pos'
does actually become exactly 180 degrees at some point of the swing.

ROB ANDERSON. SEPTEMBER 1986 

00000000  0d 41 4e 49 4d 41 54 49  4f 4e 20 54 45 43 48 4e  |.ANIMATION TECHN|
00000010  49 51 55 45 53 20 57 49  54 48 20 47 52 41 56 49  |IQUES WITH GRAVI|
00000020  54 59 0d 0d 0d 57 68 65  6e 20 77 72 69 74 69 6e  |TY...When writin|
00000030  67 20 61 20 67 61 6d 65  2c 20 6f 72 20 61 6e 79  |g a game, or any|
00000040  20 70 72 6f 67 72 61 6d  20 69 6e 20 77 68 69 63  | program in whic|
00000050  68 20 6d 6f 76 65 6d 65  6e 74 20 6f 66 20 67 72  |h movement of gr|
00000060  61 70 68 69 63 73 20 73  70 72 69 74 65 73 0d 69  |aphics sprites.i|
00000070  73 20 69 6e 63 6f 72 70  6f 72 61 74 65 64 2c 20  |s incorporated, |
00000080  69 74 20 69 73 20 6f 66  74 65 6e 20 64 65 73 69  |it is often desi|
00000090  72 61 62 6c 65 20 74 6f  20 6d 61 6b 65 20 20 63  |rable to make  c|
000000a0  65 72 74 61 69 6e 20 63  68 61 72 61 63 74 65 72  |ertain character|
000000b0  73 20 6d 6f 76 65 0d 77  69 74 68 20 61 20 72 65  |s move.with a re|
000000c0  61 6c 69 73 6d 20 74 68  61 74 20 77 65 20 65 78  |alism that we ex|
000000d0  70 65 72 69 65 6e 63 65  20 69 6e 20 6c 69 66 65  |perience in life|
000000e0  2e 20 47 72 61 76 69 74  79 20 69 73 20 6f 6e 65  |. Gravity is one|
000000f0  20 6f 66 20 74 68 65 73  65 0d 72 65 61 6c 69 74  | of these.realit|
00000100  69 65 73 2c 20 61 6e 64  20 77 68 65 6e 20 69 6e  |ies, and when in|
00000110  63 6f 72 70 6f 72 61 74  65 64 20 69 6e 20 61 20  |corporated in a |
00000120  70 72 6f 67 72 61 6d 20  74 68 65 20 65 66 66 65  |program the effe|
00000130  63 74 73 20 63 61 6e 20  62 65 20 71 75 69 74 65  |cts can be quite|
00000140  0d 69 6e 74 65 72 65 73  74 69 6e 67 20 28 6f 72  |.interesting (or|
00000150  20 73 74 61 72 74 6c 69  6e 67 20 77 68 65 6e 20  | startling when |
00000160  74 68 65 79 20 20 67 6f  20 77 72 6f 6e 67 21 29  |they  go wrong!)|
00000170  2e 0d 0d 47 72 61 76 69  74 79 20 69 73 20 61 20  |...Gravity is a |
00000180  66 6f 72 63 65 20 77 68  69 63 68 20 61 63 74 73  |force which acts|
00000190  20 6f 6e 20 61 6c 6c 20  62 6f 64 69 65 73 20 74  | on all bodies t|
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000001b0  75 74 20 62 65 63 61 75  73 65 20 61 0d 73 70 72  |ut because a.spr|
000001c0  69 74 65 20 68 61 73 20  6e 6f 20 6d 61 73 73 20  |ite has no mass |
000001d0  28 69 74 27 73 20 62 65  69 6e 67 20 68 65 6c 64  |(it's being held|
000001e0  20 69 6e 20 63 6f 6d 70  75 74 65 72 20 6d 65 6d  | in computer mem|
000001f0  6f 72 79 29 20 77 65 20  68 61 76 65 20 74 6f 0d  |ory) we have to.|
00000200  69 6e 74 72 6f 64 75 63  65 20 61 6e 20 61 72 74  |introduce an art|
00000210  69 66 69 63 69 61 6c 20  67 72 61 76 69 74 79 20  |ificial gravity |
00000220  69 6e 74 6f 20 74 68 65  20 61 6e 69 6d 61 74 69  |into the animati|
00000230  6f 6e 2e 20 54 68 65 20  69 6e 74 72 6f 64 75 63  |on. The introduc|
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00000250  68 61 73 20 6d 61 6e 79  20 61 70 70 6c 69 63 61  |has many applica|
00000260  74 69 6f 6e 73 20 77 68  69 63 68 20 69 6e 63 6c  |tions which incl|
00000270  75 64 65 20 74 68 65 20  6d 6f 76 65 6d 65 6e 74  |ude the movement|
00000280  20 6f 66 20 61 20 62 6f  75 6e 63 69 6e 67 0d 62  | of a bouncing.b|
00000290  61 6c 6c 2c 20 74 68 65  20 73 6c 6f 77 69 6e 67  |all, the slowing|
000002a0  20 64 6f 77 6e 20 61 6e  64 20 74 68 65 6e 20 73  | down and then s|
000002b0  70 65 65 64 69 6e 67 20  75 70 20 6f 66 20 61 20  |peeding up of a |
000002c0  73 70 72 69 74 65 20 77  68 65 6e 20 69 74 20 6a  |sprite when it j|
000002d0  75 6d 70 73 20 75 70 0d  61 6e 64 20 74 68 65 6e  |umps up.and then|
000002e0  20 64 6f 77 6e 20 69 6e  20 61 20 67 61 6d 65 2c  | down in a game,|
000002f0  20 6f 72 20 74 68 65 20  72 65 61 6c 69 73 74 69  | or the realisti|
00000300  63 20 73 77 69 6e 67 20  6f 66 20 61 20 72 6f 70  |c swing of a rop|
00000310  65 20 69 6e 20 67 61 6d  65 73 20 6c 69 6b 65 0d  |e in games like.|
00000320  48 55 4e 43 48 42 41 43  4b 2e 20 57 68 65 6e 20  |HUNCHBACK. When |
00000330  61 70 70 6c 69 65 64 20  69 6e 20 61 20 68 6f 72  |applied in a hor|
00000340  69 7a 6f 6e 74 61 6c 20  70 6c 61 6e 65 20 74 68  |izontal plane th|
00000350  65 20 6d 65 74 68 6f 64  73 20 64 65 73 63 69 62  |e methods descib|
00000360  65 64 20 63 61 6e 0d 61  6c 73 6f 20 62 65 20 75  |ed can.also be u|
00000370  73 65 64 20 74 6f 20 73  69 6d 75 6c 61 74 65 2c  |sed to simulate,|
00000380  20 66 6f 72 20 65 78 61  6d 70 6c 65 2c 20 73 6e  | for example, sn|
00000390  6f 6f 6b 65 72 20 62 61  6c 6c 73 20 6f 6e 20 61  |ooker balls on a|
000003a0  20 74 61 62 6c 65 20 74  68 61 74 20 68 61 73 0d  | table that has.|
000003b0  66 72 69 63 74 69 6f 6e  2e 0d 0d 50 52 4f 4a 45  |friction...PROJE|
000003c0  43 54 49 4c 45 20 4d 4f  56 45 4d 45 4e 54 0d 0d  |CTILE MOVEMENT..|
000003d0  50 72 6f 67 72 61 6d 20  31 20 69 73 20 61 20 67  |Program 1 is a g|
000003e0  65 6e 65 72 61 6c 20 70  75 72 70 6f 73 65 20 62  |eneral purpose b|
000003f0  61 6c 6c 20 74 68 72 6f  77 69 6e 67 20 70 72 6f  |all throwing pro|
00000400  67 72 61 6d 20 28 21 29  2e 20 56 61 72 69 6f 75  |gram (!). Variou|
00000410  73 0d 70 61 72 61 6d 65  74 65 72 73 20 63 61 6e  |s.parameters can|
00000420  20 62 65 20 63 68 61 6e  67 65 64 20 73 6f 20 74  | be changed so t|
00000430  68 61 74 20 74 68 65 72  65 20 65 66 66 65 63 74  |hat there effect|
00000440  73 20 75 70 6f 6e 20 74  68 65 20 62 61 6c 6c 73  |s upon the balls|
00000450  20 6d 6f 76 65 6d 65 6e  74 0d 63 61 6e 20 62 65  | movement.can be|
00000460  20 73 65 65 6e 20 62 79  20 61 6c 74 65 72 69 6e  | seen by alterin|
00000470  67 20 39 20 64 69 66 66  65 72 65 6e 74 20 76 61  |g 9 different va|
00000480  6c 75 65 73 2e 20 54 68  65 73 65 20 70 61 72 61  |lues. These para|
00000490  6d 65 74 65 72 73 20 61  72 65 20 61 6c 74 65 72  |meters are alter|
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000004b0  68 65 20 63 6f 72 72 65  73 70 6f 6e 64 69 6e 67  |he corresponding|
000004c0  20 6e 75 6d 62 65 72 20  61 6e 64 20 74 68 65 6e  | number and then|
000004d0  2c 20 69 66 20 61 20 6e  75 6d 62 65 72 20 69 73  |, if a number is|
000004e0  20 72 65 71 75 69 72 65  64 20 74 68 65 0d 75 73  | required the.us|
000004f0  65 72 20 69 73 20 70 72  6f 6d 70 74 65 64 20 74  |er is prompted t|
00000500  6f 20 65 6e 74 65 72 20  61 20 76 61 6c 75 65 2c  |o enter a value,|
00000510  20 6f 74 68 65 72 77 69  73 65 20 69 66 20 74 68  | otherwise if th|
00000520  65 20 6f 70 74 69 6f 6e  20 69 73 20 6f 66 20 61  |e option is of a|
00000530  20 59 45 53 2f 4e 4f 0d  74 79 70 65 2c 20 74 68  | YES/NO.type, th|
00000540  65 20 76 61 6c 75 65 20  77 69 6c 6c 20 74 6f 67  |e value will tog|
00000550  67 6c 65 2e 20 54 68 65  20 64 65 66 61 75 6c 74  |gle. The default|
00000560  20 76 61 6c 75 65 73 20  61 72 65 20 73 75 63 68  | values are such|
00000570  20 74 68 61 74 20 74 68  65 20 62 61 6c 6c 0d 77  | that the ball.w|
00000580  69 6c 6c 20 62 65 20 74  68 72 6f 77 6e 20 75 70  |ill be thrown up|
00000590  20 69 6e 20 74 68 65 20  27 61 69 72 27 20 61 6e  | in the 'air' an|
000005a0  64 20 74 68 65 6e 20 72  65 74 75 72 6e 20 74 6f  |d then return to|
000005b0  20 67 72 6f 75 6e 64 2e  20 54 68 65 20 61 66 74  | ground. The aft|
000005c0  65 72 20 69 6d 61 67 65  0d 6f 66 20 74 68 65 20  |er image.of the |
000005d0  62 61 6c 6c 20 63 61 6e  20 62 65 20 74 75 72 6e  |ball can be turn|
000005e0  65 64 20 6f 66 66 20 75  73 69 6e 67 20 6f 70 74  |ed off using opt|
000005f0  69 6f 6e 20 38 2c 20 61  73 20 63 61 6e 20 74 68  |ion 8, as can th|
00000600  65 20 6c 69 6e 65 20 72  65 66 65 72 65 6e 63 65  |e line reference|
00000610  73 2e 0d 41 20 66 65 77  20 70 6f 69 6e 74 73 20  |s..A few points |
00000620  6d 61 79 20 62 65 20 6e  6f 74 65 64 20 3a 0d 0d  |may be noted :..|
00000630  31 2e 20 54 68 65 20 58  20 76 65 6c 6f 63 69 74  |1. The X velocit|
00000640  79 20 61 6e 64 20 74 68  65 20 59 20 76 65 6c 6f  |y and the Y velo|
00000650  63 69 74 79 20 61 72 65  20 54 4f 54 41 4c 4c 59  |city are TOTALLY|
00000660  20 69 6e 64 65 70 65 6e  64 61 6e 74 20 6f 66 20  | independant of |
00000670  65 61 63 68 0d 6f 74 68  65 72 2e 0d 0d 32 2e 20  |each.other...2. |
00000680  54 68 65 20 58 20 76 65  6c 6f 63 69 74 79 20 72  |The X velocity r|
00000690  65 6d 61 69 6e 73 20 63  6f 6e 73 74 61 6e 74 20  |emains constant |
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000006d0  72 61 74 69 6f 6e 20 28  67 72 61 76 69 74 79 29  |ration (gravity)|
000006e0  20 61 63 74 69 6e 67 20  69 6e 20 74 68 65 20 58  | acting in the X|
000006f0  20 70 6c 61 6e 65 2e 0d  0d 33 2e 20 54 68 65 20  | plane...3. The |
00000700  69 6e 69 74 69 61 6c 20  59 20 76 65 6c 6f 63 69  |initial Y veloci|
00000710  74 79 20 77 69 6c 6c 2c  20 69 6e 20 67 65 6e 65  |ty will, in gene|
00000720  72 61 6c 2c 20 62 65 20  6c 61 72 67 65 20 74 6f  |ral, be large to|
00000730  20 73 74 61 72 74 20 77  69 74 68 2c 20 72 65 64  | start with, red|
00000740  75 63 65 0d 61 73 20 74  68 65 20 62 61 6c 6c 20  |uce.as the ball |
00000750  67 65 74 73 20 68 69 67  68 65 72 20 75 6e 74 69  |gets higher unti|
00000760  6c 20 69 74 20 73 74 6f  70 73 2c 20 61 6e 64 20  |l it stops, and |
00000770  74 68 65 6e 20 69 74 20  77 69 6c 6c 20 62 65 67  |then it will beg|
00000780  69 6e 20 74 6f 20 67 61  69 6e 0d 73 70 65 65 64  |in to gain.speed|
00000790  20 61 73 20 69 74 20 66  61 6c 6c 73 20 62 61 63  | as it falls bac|
000007a0  6b 20 64 6f 77 6e 20 74  6f 20 74 68 65 20 67 72  |k down to the gr|
000007b0  6f 75 6e 64 2e 0d 0d 34  2e 20 42 79 20 61 6c 74  |ound...4. By alt|
000007c0  65 72 69 6e 67 20 74 68  65 20 58 20 76 65 6c 6f  |ering the X velo|
000007d0  63 69 74 79 20 69 74 20  6d 61 79 20 62 65 20 73  |city it may be s|
000007e0  65 65 6e 20 74 68 61 74  20 74 68 65 20 72 61 6e  |een that the ran|
000007f0  67 65 20 6f 66 20 74 68  65 20 62 61 6c 6c 20 69  |ge of the ball i|
00000800  73 0d 69 6e 63 72 65 61  73 65 64 20 62 65 63 61  |s.increased beca|
00000810  75 73 65 20 74 68 65 20  62 61 6c 6c 20 63 61 6e  |use the ball can|
00000820  20 74 72 61 76 65 6c 20  66 75 72 74 68 65 72 20  | travel further |
00000830  74 6f 20 74 68 65 20 72  69 67 68 74 20 62 65 66  |to the right bef|
00000840  6f 72 65 20 69 74 0d 72  65 74 75 72 6e 73 20 74  |ore it.returns t|
00000850  6f 20 74 68 65 20 67 72  6f 75 6e 64 2e 0d 0d 54  |o the ground...T|
00000860  68 65 20 62 65 73 74 20  77 61 79 20 74 6f 20 75  |he best way to u|
00000870  6e 64 65 72 73 74 61 6e  64 20 77 68 61 74 20 69  |nderstand what i|
00000880  73 20 67 6f 69 6e 67 20  6f 6e 20 69 73 20 74 6f  |s going on is to|
00000890  20 73 69 6d 70 6c 79 20  70 6c 61 79 20 77 69 74  | simply play wit|
000008a0  68 20 74 68 65 0d 70 72  6f 67 72 61 6d 2e 0d 0d  |h the.program...|
000008b0  48 4f 57 20 47 52 41 56  49 54 59 20 49 53 20 53  |HOW GRAVITY IS S|
000008c0  49 4d 55 4c 41 54 45 44  0d 0d 54 68 65 20 6d 6f  |IMULATED..The mo|
000008d0  76 65 6d 65 6e 74 20 6f  66 20 74 68 65 20 62 61  |vement of the ba|
000008e0  6c 6c 20 75 73 65 73 20  73 69 6d 70 6c 65 20 61  |ll uses simple a|
000008f0  6e 69 6d 61 74 69 6f 6e  20 74 65 63 68 6e 69 71  |nimation techniq|
00000900  75 65 73 2c 20 61 6e 64  20 74 68 65 0d 69 6e 74  |ues, and the.int|
00000910  72 6f 64 75 63 74 69 6f  6e 20 6f 66 20 67 72 61  |roduction of gra|
00000920  76 69 74 79 20 69 73 20  61 63 68 69 65 76 65 64  |vity is achieved|
00000930  20 6f 6e 20 6c 69 6e 65  20 33 37 30 2e 20 54 68  | on line 370. Th|
00000940  65 20 76 61 72 69 61 62  6c 65 20 27 79 76 65 6c  |e variable 'yvel|
00000950  25 27 0d 68 6f 6c 64 73  20 61 20 76 61 6c 75 65  |%'.holds a value|
00000960  20 74 68 61 74 20 72 65  70 72 65 73 65 6e 74 73  | that represents|
00000970  20 74 68 65 20 59 20 76  65 6c 6f 63 69 74 79 20  | the Y velocity |
00000980  6f 66 20 74 68 65 20 62  61 6c 6c 20 69 2e 65 2e  |of the ball i.e.|
00000990  20 69 74 20 69 73 20 74  68 65 0d 6e 75 6d 62 65  | it is the.numbe|
000009a0  72 20 6f 66 20 70 69 78  69 6c 73 20 77 68 69 63  |r of pixils whic|
000009b0  68 20 61 72 65 20 6a 75  6d 70 65 64 20 64 75 72  |h are jumped dur|
000009c0  69 6e 67 20 73 75 62 73  65 71 75 65 6e 74 20 70  |ing subsequent p|
000009d0  6c 6f 74 74 69 6e 67 20  6f 66 20 74 68 65 20 62  |lotting of the b|
000009e0  61 6c 6c 2e 0d 54 68 75  73 20 69 6e 69 74 69 61  |all..Thus initia|
000009f0  6c 6c 79 20 27 79 76 65  6c 25 27 20 69 73 20 6c  |lly 'yvel%' is l|
00000a00  61 72 67 65 20 73 6f 20  74 68 61 74 20 74 68 65  |arge so that the|
00000a10  20 62 61 6c 6c 20 61 70  70 65 61 72 73 20 74 6f  | ball appears to|
00000a20  20 6d 6f 76 65 20 66 61  73 74 2c 20 62 75 74 0d  | move fast, but.|
00000a30  74 68 69 73 20 76 61 6c  75 65 20 69 73 20 73 6c  |this value is sl|
00000a40  6f 77 6c 79 20 64 65 63  72 65 61 73 65 64 20 62  |owly decreased b|
00000a50  79 20 73 75 62 74 72 61  63 74 69 6e 67 20 61 20  |y subtracting a |
00000a60  76 61 6c 75 65 20 68 65  6c 64 20 69 6e 20 27 67  |value held in 'g|
00000a70  72 61 76 69 74 79 27 20  73 6f 0d 74 68 61 74 20  |ravity' so.that |
00000a80  74 68 65 20 62 61 6c 6c  20 61 70 70 65 61 72 73  |the ball appears|
00000a90  20 74 6f 20 73 6c 6f 77  20 64 6f 77 6e 20 61 73  | to slow down as|
00000aa0  20 69 74 20 67 65 74 73  20 68 69 67 68 65 72 2e  | it gets higher.|
00000ab0  20 45 76 65 6e 74 75 61  6c 6c 79 20 27 79 76 65  | Eventually 'yve|
00000ac0  6c 25 27 0d 77 69 6c 6c  20 62 65 63 6f 6d 65 20  |l%'.will become |
00000ad0  7a 65 72 6f 2c 20 61 66  74 65 72 77 68 69 63 68  |zero, afterwhich|
00000ae0  20 69 74 20 77 69 6c 6c  20 62 65 63 6f 6d 65 20  | it will become |
00000af0  6e 65 67 61 74 69 76 65  20 28 69 2e 65 2e 20 61  |negative (i.e. a|
00000b00  20 2d 76 65 0d 76 65 6c  6f 63 69 74 79 29 2c 20  | -ve.velocity), |
00000b10  73 6f 20 74 68 65 20 62  61 6c 6c 20 77 69 6c 6c  |so the ball will|
00000b20  20 73 6c 6f 77 6c 79 20  62 65 67 69 6e 20 74 6f  | slowly begin to|
00000b30  20 6d 6f 76 65 20 69 6e  20 74 68 65 20 6f 70 70  | move in the opp|
00000b40  6f 73 69 74 65 20 64 69  72 65 63 74 69 6f 6e 0d  |osite direction.|
00000b50  61 6e 64 20 74 68 65 6e  20 67 61 74 68 65 72 20  |and then gather |
00000b60  73 70 65 65 64 2e 20 54  68 65 20 76 61 72 69 61  |speed. The varia|
00000b70  62 6c 65 20 27 67 72 61  76 69 74 79 27 20 72 65  |ble 'gravity' re|
00000b80  70 72 65 73 65 6e 74 73  20 74 68 65 20 72 61 74  |presents the rat|
00000b90  65 20 6f 66 0d 63 68 61  6e 67 65 20 6f 66 20 74  |e of.change of t|
00000ba0  68 65 20 59 20 76 65 6c  6f 63 69 74 79 20 28 69  |he Y velocity (i|
00000bb0  74 73 20 61 63 63 65 6c  65 72 61 74 69 6f 6e 2c  |ts acceleration,|
00000bc0  20 6f 72 20 27 74 68 65  20 67 72 61 76 69 74 61  | or 'the gravita|
00000bd0  74 69 6f 6e 61 6c 20 66  69 65 6c 64 0d 73 74 72  |tional field.str|
00000be0  65 6e 67 74 68 27 29 2e  0d 0d 43 49 52 43 55 4c  |ength')...CIRCUL|
00000bf0  41 52 20 4d 4f 54 49 4f  4e 20 41 4e 44 20 47 52  |AR MOTION AND GR|
00000c00  41 56 49 54 59 20 2d 20  74 68 65 20 70 65 6e 64  |AVITY - the pend|
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00000d90  6e 20 6f 73 63 69 6c 6c  61 74 69 6f 6e 73 20 6f  |n oscillations o|
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00000e20  20 70 72 6f 67 72 61 6d  20 31 2c 20 69 74 20 69  | program 1, it i|
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00000e90  6d 20 32 20 73 69 6d 75  6c 61 74 65 73 20 74 72  |m 2 simulates tr|
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00000f00  39 30 20 64 65 67 72 65  65 73 20 6f 75 74 20 6f  |90 degrees out o|
00000f10  66 20 70 68 61 73 65 0d  2d 20 64 6f 6e 27 74 20  |f phase.- don't |
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00000f70  6e 67 20 61 63 68 69 65  76 65 64 20 74 68 69 73  |ng achieved this|
00000f80  2c 20 74 68 65 20 69 6e  69 74 69 61 6c 20 70 6f  |, the initial po|
00000f90  73 69 74 69 6f 6e 20 6f  66 20 74 68 65 20 62 6f  |sition of the bo|
00000fa0  62 20 69 73 0d 64 65 63  69 64 65 64 20 61 6e 64  |b is.decided and|
00000fb0  20 74 68 65 6e 20 74 68  65 20 70 65 6e 64 75 6c  | then the pendul|
00000fc0  75 6d 20 69 73 20 61 6c  6c 6f 77 65 64 20 74 6f  |um is allowed to|
00000fd0  20 72 6f 74 61 74 65 20  69 6e 20 61 20 63 69 72  | rotate in a cir|
00000fe0  63 75 6c 61 72 20 70 61  74 68 0d 69 6e 69 74 69  |cular path.initi|
00000ff0  61 6c 6c 79 20 6d 6f 76  69 6e 67 20 64 6f 77 6e  |ally moving down|
00001000  77 61 72 64 73 2e 20 54  68 65 20 70 72 6f 63 65  |wards. The proce|
00001010  64 75 72 65 20 27 78 79  27 20 63 61 6c 63 75 6c  |dure 'xy' calcul|
00001020  61 74 65 73 20 74 68 65  20 78 2d 79 20 70 6f 73  |ates the x-y pos|
00001030  69 74 69 6f 6e 0d 6f 66  20 74 68 65 20 62 6f 62  |ition.of the bob|
00001040  20 61 74 20 61 6e 79 20  61 6e 67 6c 65 20 28 68  | at any angle (h|
00001050  65 6c 64 20 69 6e 20 27  70 6f 73 27 29 20 6f 66  |eld in 'pos') of|
00001060  20 74 68 65 20 63 69 72  63 6c 65 2e 20 54 68 75  | the circle. Thu|
00001070  73 2c 20 74 6f 20 6f 62  74 61 69 6e 0d 63 69 72  |s, to obtain.cir|
00001080  63 75 6c 65 72 20 6d 6f  74 69 6f 6e 20 69 74 20  |culer motion it |
00001090  69 73 20 73 69 6d 70 6c  79 20 61 20 6d 61 74 74  |is simply a matt|
000010a0  65 72 20 6f 66 20 67 69  76 69 6e 67 20 27 70 6f  |er of giving 'po|
000010b0  73 27 20 63 6f 6e 73 65  63 75 74 69 76 65 20 76  |s' consecutive v|
000010c0  61 6c 75 65 73 0d 66 72  6f 6d 20 30 20 74 6f 20  |alues.from 0 to |
000010d0  33 36 30 20 64 67 72 65  65 73 20 61 6e 64 20 70  |360 dgrees and p|
000010e0  6c 6f 74 74 69 6e 67 20  65 61 63 68 20 78 2d 79  |lotting each x-y|
000010f0  20 63 6f 72 72 64 69 6e  61 74 65 20 66 6f 72 20  | corrdinate for |
00001100  74 68 65 0d 63 6f 72 72  65 73 70 6f 6e 64 69 6e  |the.correspondin|
00001110  67 20 76 61 6c 75 65 20  6f 66 20 27 70 6f 73 27  |g value of 'pos'|
00001120  2e 0d 0d 48 6f 77 65 76  65 72 2c 20 77 65 20 77  |...However, we w|
00001130  61 6e 74 20 74 68 65 20  70 65 6e 64 75 6c 75 6d  |ant the pendulum|
00001140  20 74 6f 20 73 74 61 72  74 20 66 72 6f 6d 20 61  | to start from a|
00001150  20 68 6f 72 69 7a 6f 6e  74 61 6c 20 70 6f 73 69  | horizontal posi|
00001160  74 69 6f 6e 2c 20 73 6f  20 74 68 65 0d 69 6e 69  |tion, so the.ini|
00001170  74 69 61 6c 20 76 61 6c  75 65 20 6f 66 20 27 70  |tial value of 'p|
00001180  6f 73 27 20 69 73 20 73  65 74 20 74 6f 20 39 30  |os' is set to 90|
00001190  20 64 65 67 72 65 65 73  20 28 6c 69 6e 65 20 32  | degrees (line 2|
000011a0  34 30 29 2e 20 54 68 65  20 73 70 65 65 64 20 6f  |40). The speed o|
000011b0  66 20 74 68 65 0d 62 6f  62 20 69 73 20 64 65 74  |f the.bob is det|
000011c0  65 72 6d 69 6e 65 64 20  62 79 20 74 68 65 20 73  |ermined by the s|
000011d0  74 65 70 20 72 61 74 65  20 6f 66 20 74 68 65 20  |tep rate of the |
000011e0  61 6e 67 6c 65 20 74 68  72 6f 75 67 68 20 77 68  |angle through wh|
000011f0  69 63 68 20 74 68 65 20  62 6f 62 0d 74 75 72 6e  |ich the bob.turn|
00001200  73 20 28 69 2e 65 2e 20  74 68 65 20 72 61 74 65  |s (i.e. the rate|
00001210  20 6f 66 20 63 68 61 6e  67 65 20 6f 66 20 27 70  | of change of 'p|
00001220  6f 73 27 29 2e 20 54 68  69 73 20 73 74 65 70 20  |os'). This step |
00001230  72 61 74 65 20 69 73 20  68 65 6c 64 20 69 6e 20  |rate is held in |
00001240  74 68 65 0d 76 61 72 69  61 62 6c 65 20 27 76 65  |the.variable 've|
00001250  6c 27 20 73 6f 20 69 66  2c 20 66 6f 72 20 65 78  |l' so if, for ex|
00001260  61 6d 70 6c 65 2c 20 27  76 65 6c 27 20 3d 20 31  |ample, 'vel' = 1|
00001270  30 2c 20 74 68 65 6e 20  74 68 65 20 62 6f 62 20  |0, then the bob |
00001280  77 6f 75 6c 64 20 6d 6f  76 65 20 69 6e 0d 61 20  |would move in.a |
00001290  63 69 72 63 6c 65 20 69  6e 20 6a 75 6d 70 73 20  |circle in jumps |
000012a0  6f 66 20 31 30 20 64 65  67 72 65 65 73 2e 20 49  |of 10 degrees. I|
000012b0  74 20 69 73 20 6e 6f 77  20 61 20 73 69 6d 70 6c  |t is now a simpl|
000012c0  65 20 6d 61 74 74 65 72  20 74 6f 20 69 6e 74 72  |e matter to intr|
000012d0  6f 64 75 63 65 0d 67 72  61 76 69 74 79 20 28 61  |oduce.gravity (a|
000012e0  73 20 77 65 20 64 69 64  20 69 6e 20 70 72 6f 67  |s we did in prog|
000012f0  72 61 6d 20 31 20 29 20  62 79 20 69 6e 63 72 65  |ram 1 ) by incre|
00001300  61 73 69 6e 67 20 74 68  65 20 76 65 6c 6f 63 69  |asing the veloci|
00001310  74 79 20 6f 66 20 74 68  65 20 62 6f 62 20 61 73  |ty of the bob as|
00001320  0d 69 74 20 66 61 6c 6c  73 2e 20 54 68 69 73 20  |.it falls. This |
00001330  69 73 20 61 63 68 69 65  76 65 64 20 6f 6e 20 6c  |is achieved on l|
00001340  69 6e 65 20 33 36 30 2e  0d 0d 54 68 65 72 65 20  |ine 360...There |
00001350  69 73 2c 20 68 6f 77 65  76 65 72 2c 20 6f 6e 65  |is, however, one|
00001360  20 70 72 6f 62 6c 65 6d  2e 20 42 65 63 61 75 73  | problem. Becaus|
00001370  65 20 74 68 65 20 76 65  6c 6f 63 69 74 79 20 69  |e the velocity i|
00001380  73 20 63 6f 6e 73 74 61  6e 74 6c 79 0d 69 6e 63  |s constantly.inc|
00001390  72 65 61 73 69 6e 67 20  66 72 6f 6d 20 61 20 70  |reasing from a p|
000013a0  6f 73 69 74 69 76 65 20  76 61 6c 75 65 20 61 73  |ositive value as|
000013b0  20 69 74 20 73 77 69 6e  67 73 20 64 6f 77 6e 77  | it swings downw|
000013c0  61 72 64 73 2c 20 69 74  20 77 69 6c 6c 20 6e 65  |ards, it will ne|
000013d0  76 65 72 0d 72 65 61 63  68 20 61 20 74 75 72 6e  |ver.reach a turn|
000013e0  69 6e 67 20 70 6f 69 6e  74 20 61 74 20 77 68 69  |ing point at whi|
000013f0  63 68 20 74 68 65 20 76  65 6c 6f 63 69 74 79 20  |ch the velocity |
00001400  77 69 6c 6c 20 62 65 67  69 6e 20 74 6f 20 67 65  |will begin to ge|
00001410  74 20 73 6c 6f 77 65 72  2c 20 73 6f 0d 74 68 65  |t slower, so.the|
00001420  20 72 65 73 75 6c 74 20  77 69 6c 6c 20 62 65 20  | result will be |
00001430  61 20 63 6f 6e 73 74 61  6e 74 6c 79 20 61 63 63  |a constantly acc|
00001440  65 6c 65 72 61 74 69 6e  67 20 62 6f 62 20 6d 6f  |elerating bob mo|
00001450  76 69 6e 67 20 69 6e 20  61 20 63 69 72 63 6c 65  |ving in a circle|
00001460  2e 20 54 68 75 73 0d 77  65 20 6d 75 73 74 20 70  |. Thus.we must p|
00001470  75 74 20 69 6e 20 61 20  63 6f 6e 64 69 74 69 6f  |ut in a conditio|
00001480  6e 20 77 68 65 72 65 62  79 20 6f 6e 63 65 20 74  |n whereby once t|
00001490  68 65 20 62 6f 62 20 72  65 61 63 68 65 73 20 61  |he bob reaches a|
000014a0  20 63 65 72 74 61 69 6e  0d 76 65 6c 6f 63 69 74  | certain.velocit|
000014b0  79 2c 20 74 68 65 20 73  69 67 6e 20 6f 66 20 74  |y, the sign of t|
000014c0  68 65 20 67 72 61 76 69  74 79 20 69 73 20 63 68  |he gravity is ch|
000014d0  61 6e 67 65 64 20 72 65  73 75 6c 74 69 6e 67 20  |anged resulting |
000014e0  69 6e 20 74 68 65 20 62  6f 62 20 73 6c 6f 77 69  |in the bob slowi|
000014f0  6e 67 0d 64 6f 77 6e 20  75 6e 74 69 6c 20 69 74  |ng.down until it|
00001500  20 65 76 65 6e 74 75 61  6c 6c 79 20 73 74 6f 70  | eventually stop|
00001510  73 20 61 74 20 74 68 65  20 74 6f 70 20 6f 66 20  |s at the top of |
00001520  69 74 73 20 73 77 69 6e  67 20 28 73 65 65 20 6c  |its swing (see l|
00001530  69 6e 65 73 0d 33 34 30  2c 33 35 30 29 2e 20 41  |ines.340,350). A|
00001540  6c 74 65 72 6e 61 74 69  76 65 6c 79 2c 20 74 68  |lternatively, th|
00001550  65 20 73 69 67 6e 20 6f  66 20 27 67 72 61 76 69  |e sign of 'gravi|
00001560  74 79 27 20 63 6f 75 6c  64 20 62 65 20 73 77 61  |ty' could be swa|
00001570  70 70 65 64 20 77 68 65  6e 20 74 68 65 0d 76 61  |pped when the.va|
00001580  6c 75 65 20 6f 66 20 27  70 6f 73 27 20 65 71 75  |lue of 'pos' equ|
00001590  61 6c 65 64 20 31 38 30  20 64 65 67 72 65 65 73  |aled 180 degrees|
000015a0  2c 20 62 75 74 20 79 6f  75 20 68 61 76 65 20 74  |, but you have t|
000015b0  6f 20 6d 61 6b 65 20 73  75 72 65 20 74 68 61 74  |o make sure that|
000015c0  20 27 70 6f 73 27 0d 64  6f 65 73 20 61 63 74 75  | 'pos'.does actu|
000015d0  61 6c 6c 79 20 62 65 63  6f 6d 65 20 65 78 61 63  |ally become exac|
000015e0  74 6c 79 20 31 38 30 20  64 65 67 72 65 65 73 20  |tly 180 degrees |
000015f0  61 74 20 73 6f 6d 65 20  70 6f 69 6e 74 20 6f 66  |at some point of|
00001600  20 74 68 65 20 73 77 69  6e 67 2e 0d 0d 52 4f 42  | the swing...ROB|
00001610  20 41 4e 44 45 52 53 4f  4e 2e 20 53 45 50 54 45  | ANDERSON. SEPTE|
00001620  4d 42 45 52 20 31 39 38  36 20 0d                 |MBER 1986 .|
0000162b
26-09-88/GRAVTXT.m0
26-09-88/GRAVTXT.m1
26-09-88/GRAVTXT.m2
26-09-88/GRAVTXT.m4
26-09-88/GRAVTXT.m5

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