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|00000000| 00 07 66 61 6c 6c 69 6e | 67 00 00 00 00 00 00 00 |..fallin|g.......|
|00000010| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000020| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000030| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000040| 00 54 45 58 54 4d 50 61 | 64 01 00 00 00 00 00 00 |.TEXTMPa|d.......|
|00000050| 00 00 00 00 00 04 9c 00 | 00 01 56 aa 19 74 77 aa |........|..V..tw.|
|00000060| 19 74 77 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |.tw.....|........|
|00000070| 00 00 00 00 00 00 00 00 | 00 00 81 81 96 ab 00 00 |........|........|
|00000080| 2d 2d 54 68 69 73 20 65 | 78 61 6d 70 6c 65 20 73 |--This e|xample s|
|00000090| 69 6d 75 6c 61 74 65 73 | 20 61 20 66 61 6c 6c 69 |imulates| a falli|
|000000a0| 6e 67 20 6f 62 6a 65 63 | 74 20 77 69 74 68 20 61 |ng objec|t with a|
|000000b0| 69 72 20 66 72 69 63 74 | 69 6f 6e 20 70 72 6f 70 |ir frict|ion prop|
|000000c0| 6f 72 74 69 6f 6e 61 6c | 20 74 6f 20 74 68 65 20 |ortional| to the |
|000000d0| 73 71 75 61 72 65 20 6f | 66 20 69 74 73 20 76 65 |square o|f its ve|
|000000e0| 6c 6f 63 69 74 79 2e 20 | 54 68 65 20 74 65 63 68 |locity. |The tech|
|000000f0| 6e 69 71 75 65 20 75 73 | 65 64 20 69 73 20 69 6e |nique us|ed is in|
|00000100| 65 66 66 69 63 69 65 6e | 74 20 63 6f 6d 70 61 72 |efficien|t compar|
|00000110| 65 64 20 77 69 74 68 20 | 66 61 6e 63 69 65 72 20 |ed with |fancier |
|00000120| 6d 65 74 68 6f 64 73 20 | 66 6f 72 20 6e 75 6d 65 |methods |for nume|
|00000130| 72 69 63 61 6c 20 73 6f | 6c 75 74 69 6f 6e 73 20 |rical so|lutions |
|00000140| 6f 66 20 64 69 66 66 65 | 72 65 6e 74 69 61 6c 20 |of diffe|rential |
|00000150| 65 71 75 61 74 69 6f 6e | 73 2c 20 62 75 74 20 69 |equation|s, but i|
|00000160| 74 20 69 73 20 76 65 72 | 79 20 65 61 73 79 20 74 |t is ver|y easy t|
|00000170| 6f 20 70 72 6f 67 72 61 | 6d 2e 20 53 65 65 20 65 |o progra|m. See e|
|00000180| 78 61 6d 70 6c 65 20 22 | 70 72 6f 6a 65 63 74 69 |xample "|projecti|
|00000190| 6c 65 22 20 66 6f 72 20 | 61 20 73 69 6d 69 6c 61 |le" for |a simila|
|000001a0| 72 20 70 72 6f 62 6c 65 | 6d 20 69 6e 20 74 77 6f |r proble|m in two|
|000001b0| 20 64 69 6d 65 6e 73 69 | 6f 6e 73 2e 0d 0d 2d 2d | dimensi|ons...--|
|000001c0| 20 73 74 65 70 74 6f 28 | 30 29 20 73 65 74 73 20 | stepto(|0) sets |
|000001d0| 75 70 20 69 6e 69 74 69 | 61 6c 20 63 6f 6e 64 69 |up initi|al condi|
|000001e0| 74 69 6f 6e 73 2e 20 41 | 6e 79 20 6f 74 68 65 72 |tions. A|ny other|
|000001f0| 20 73 74 6f 70 20 74 69 | 6d 65 20 72 75 6e 73 20 | stop ti|me runs |
|00000200| 74 68 65 20 73 69 6d 75 | 6c 61 74 69 6f 6e 20 66 |the simu|lation f|
|00000210| 72 6f 6d 20 77 68 65 72 | 65 20 69 74 20 6c 65 66 |rom wher|e it lef|
|00000220| 74 20 6f 66 66 20 75 70 | 20 75 6e 74 69 6c 20 74 |t off up| until t|
|00000230| 68 65 20 6e 65 77 20 73 | 74 6f 70 20 74 69 6d 65 |he new s|top time|
|00000240| 20 28 77 69 74 68 69 6e | 20 64 74 29 2e 20 53 69 | (within| dt). Si|
|00000250| 6e 63 65 20 74 68 65 20 | 70 6c 6f 74 20 63 61 6c |nce the |plot cal|
|00000260| 6c 73 20 66 6f 72 20 70 | 6f 69 6e 74 73 20 69 6e |ls for p|oints in|
|00000270| 20 6f 72 64 65 72 2c 20 | 74 68 65 20 73 69 6d 75 | order, |the simu|
|00000280| 6c 61 74 69 6f 6e 20 64 | 6f 65 73 20 6e 6f 74 20 |lation d|oes not |
|00000290| 6e 65 65 64 20 74 6f 20 | 73 74 61 72 74 20 6f 76 |need to |start ov|
|000002a0| 65 72 20 66 6f 72 20 65 | 61 63 68 20 70 6c 6f 74 |er for e|ach plot|
|000002b0| 74 65 64 20 70 6f 69 6e | 74 2e 0d 0d 69 6e 69 74 |ted poin|t...init|
|000002c0| 20 3d 20 76 3a 3d 30 2c | 74 3a 3d 30 20 20 2d 2d | = v:=0,|t:=0  --|
|000002d0| 20 69 6e 69 74 69 61 6c | 20 63 6f 6e 64 69 74 69 | initial| conditi|
|000002e0| 6f 6e 73 0d 0d 73 74 65 | 70 74 6f 28 73 74 6f 70 |ons..ste|pto(stop|
|000002f0| 29 20 3d 20 69 6e 69 74 | 20 77 68 65 6e 20 73 74 |) = init| when st|
|00000300| 6f 70 3d 30 2c 0d 20 20 | 20 20 20 20 20 20 20 20 |op=0,.  |        |
|00000310| 20 20 20 20 20 28 76 3a | 3d 76 2b 61 2a 64 74 2c |     (v:|=v+a*dt,|
|00000320| 0d 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00000330| 20 74 3a 3d 74 2b 64 74 | 29 20 77 68 69 6c 65 20 | t:=t+dt|) while |
|00000340| 74 3c 73 74 6f 70 0d 0d | 66 3d 6d 2a 67 2d 6b 2a |t<stop..|f=m*g-k*|
|00000350| 76 5e 32 20 20 20 2d 2d | 20 67 72 61 76 69 74 79 |v^2   --| gravity|
|00000360| 20 2d 20 61 69 72 20 66 | 72 69 63 74 69 6f 6e 0d | - air f|riction.|
|00000370| 61 3d 66 2f 6d 0d 20 20 | 20 20 20 20 20 20 20 20 |a=f/m.  |        |
|00000380| 20 20 0d 67 3d 39 2e 38 | 20 20 20 20 20 20 20 20 |  .g=9.8|        |
|00000390| 2d 2d 20 61 63 63 65 6c | 6c 65 72 61 74 69 6f 6e |-- accel|leration|
|000003a0| 20 64 75 65 20 74 6f 20 | 67 72 61 76 69 74 79 0d | due to |gravity.|
|000003b0| 6b 3d 32 20 20 20 20 20 | 20 20 20 20 20 2d 2d 20 |k=2     |     -- |
|000003c0| 66 72 69 63 74 69 6f 6e | 20 63 6f 65 66 66 69 63 |friction| coeffic|
|000003d0| 69 65 6e 74 0d 6d 3d 33 | 30 30 20 20 20 20 20 20 |ient.m=3|00      |
|000003e0| 20 20 2d 2d 20 6d 61 73 | 73 20 6f 66 20 6f 62 6a |  -- mas|s of obj|
|000003f0| 65 63 74 0d 64 74 3d 2e | 30 35 20 20 20 20 20 20 |ect.dt=.|05      |
|00000400| 20 2d 2d 20 74 69 6d 65 | 20 73 74 65 70 20 66 6f | -- time| step fo|
|00000410| 72 20 73 69 6d 75 6c 61 | 74 69 6f 6e 0d 0d 76 73 |r simula|tion..vs|
|00000420| 69 6d 28 74 31 29 20 3d | 20 73 74 65 70 74 6f 28 |im(t1) =| stepto(|
|00000430| 74 31 29 2c 76 20 20 2d | 2d 72 75 6e 20 74 6f 20 |t1),v  -|-run to |
|00000440| 74 31 2c 20 72 65 74 75 | 72 6e 20 76 0d 0d 58 6d |t1, retu|rn v..Xm|
|00000450| 69 6e 3d 30 3b 20 58 6d | 61 78 3d 31 30 20 20 2d |in=0; Xm|ax=10  -|
|00000460| 2d 20 70 6c 6f 74 20 76 | 20 76 73 20 74 69 6d 65 |- plot v| vs time|
|00000470| 20 66 6f 72 20 31 30 20 | 73 65 63 0d 70 6c 6f 74 | for 10 |sec.plot|
|00000480| 20 76 73 69 6d 28 58 29 | 0d 0d 2d 2d 20 63 6f 6d | vsim(X)|..-- com|
|00000490| 70 61 72 65 20 74 6f 20 | 61 6e 61 6c 79 74 69 63 |pare to |analytic|
|000004a0| 61 6c 20 73 6f 6c 75 74 | 69 6f 6e 0d 76 61 6e 28 |al solut|ion.van(|
|000004b0| 74 29 3d 73 71 72 74 28 | 6d 2a 67 2f 6b 29 2a 74 |t)=sqrt(|m*g/k)*t|
|000004c0| 61 6e 68 28 73 71 72 74 | 28 67 2a 6b 2f 6d 29 2a |anh(sqrt|(g*k/m)*|
|000004d0| 74 29 0d 70 6c 6f 74 20 | 76 61 6e 28 58 29 0d 0d |t).plot |van(X)..|
|000004e0| 2d 2d 20 68 79 70 65 72 | 62 6f 6c 69 63 20 74 61 |-- hyper|bolic ta|
|000004f0| 6e 0d 74 61 6e 68 28 75 | 29 3d 28 65 78 70 28 75 |n.tanh(u|)=(exp(u|
|00000500| 29 2d 65 78 70 28 2d 75 | 29 29 2f 28 65 78 70 28 |)-exp(-u|))/(exp(|
|00000510| 75 29 2b 65 78 70 28 2d | 75 29 29 0d 00 00 00 00 |u)+exp(-|u)).....|
|00000520| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000530| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000540| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000550| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000560| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000570| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000580| 00 00 01 00 00 00 01 24 | 00 00 00 24 00 00 00 32 |.......$|...$...2|
|00000590| 64 74 c0 5f 58 c8 02 22 | 54 2f 29 31 01 21 22 67 |dt._X.."|T/)1.!"g|
|000005a0| 83 01 b0 68 b1 67 01 67 | 2e 7f 7e 31 32 7a 9f 51 |...h.g.g|..~12z.Q|
|000005b0| 07 66 61 6c 6c 69 6e 67 | 02 00 00 00 54 45 58 54 |.falling|....TEXT|
|000005c0| 4d 50 61 64 01 00 00 00 | 00 80 00 00 00 00 4f 52 |MPad....|......OR|
|000005d0| 00 00 54 45 58 54 4d 50 | 61 64 01 00 00 00 00 80 |..TEXTMP|ad......|
|000005e0| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|000005f0| 00 00 a8 30 0f b8 00 00 | 04 9c 00 00 01 56 9d 67 |...0....|.....V.g|
|00000600| 8e 01 03 ca 99 02 65 78 | 74 6e 5a 5b 50 01 2f 02 |......ex|tnZ[P./.|
|00000610| 7a 5f 6d 80 01 03 b2 55 | 7a 01 ff 9e 59 5b 58 01 |z_m....U|z...Y[X.|
|00000620| 3d 68 b1 01 ff a2 76 11 | fe 8c 11 42 10 01 10 2d |=h....v.|...B...-|
|00000630| 11 fe 02 03 65 00 03 92 | 0c 00 e8 02 62 00 03 8a |....e...|....b...|
|00000640| 8b 06 12 00 d2 41 32 3b | 10 06 4e fb 10 00 75 06 |.....A2;|..N...u.|
|00000650| 00 34 00 82 00 b8 02 ac | 03 62 b9 fc 6c 6c b0 8d |.4......|.b..ll..|
|00000660| be 03 70 04 a5 22 28 48 | 8d 80 01 03 58 7f 03 42 |..p.."(H|....X..B|
|00000670| 50 21 cc 0b 64 12 1b 7c | 00 01 26 86 1a 28 18 48 |P!..d..||..&..(.H|
|00000680| 00 00 00 20 00 02 00 02 | 00 02 3f f9 8e fa 35 12 |... ....|..?...5.|
|00000690| 94 e9 c8 ae 01 2c 01 23 | 00 06 00 29 00 cc 01 24 |.....,.#|...)...$|
|000006a0| 01 33 00 28 00 00 01 00 | 00 00 01 24 00 00 00 24 |.3.(....|...$...$|
|000006b0| 00 00 00 32 00 40 44 84 | 05 26 00 00 00 1c 00 32 |...2.@D.|.&.....2|
|000006c0| 00 00 50 52 65 66 00 00 | 00 0a 00 80 ff ff 00 00 |..PRef..|........|
|000006d0| 00 00 00 41 36 3c 00 00 | 00 00 00 00 00 00 00 00 |...A6<..|........|
|000006e0| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|000006f0| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
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