さまざまな言語で数値計算

ホイン法

初期値 $ x_0 $ から次の式によって, 順次 $ x_1, x_2, \dots $ を求める.

例題として, 初速 $ 250 \mathrm{km/h}, 45^\circ $ の角度で打ったボールの軌跡を, ホイン法で計算する
(空気抵抗係数を 0.01 で計算)

重力による鉛直方向の減速分は, 重力加速度を $g$, 時間を $t$ とすると,

空気抵抗による水平方向の減速分は,速度を $v$, 速度の水平方向成分を $vx$, 空気抵抗係数を $k$ とすると,

同様に, 鉛直方向の減速分は, 速度の鉛直方向成分を $vy$ とすると,

VBScript

Option Explicit

Private Const PI = 3.14159265359

'重力加速度
Private Const g = -9.8
'空気抵抗係数
Private Const k = -0.01
'時間間隔(秒)
Private Const h = 0.01

'角度
Private Const degree = 45
Private radian: radian = degree * PI / 180.0
'初速 250 km/h -> 秒速に変換
Private v: v = 250 * 1000 \ 3600
'水平方向の速度
Private vx(): ReDim vx(2)
vx(0) = v * Cos(radian)
'鉛直方向の速度
Private vy(): ReDim vy(2)
vy(0) = v * Sin(radian)
'経過秒数
Private t: t = 0.0
'位置
Private x(): ReDim x(2)
x(0) = 0.0
Private y(): ReDim y(2)
y(0) = 0.0

'空気抵抗による水平方向の減速分
Private Function fx(ByVal vx, ByVal vy)
    fx = k * Sqr(vx * vx + vy * vy) * vx
End Function

'重力と空気抵抗による鉛直方向の減速分
Private Function fy(ByVal vx, ByVal vy)
    fy = g + (k * Sqr(vx * vx + vy * vy) * vy)
End Function

'Heun法
Dim i: i = 1
Do While (y(0) >= 0.0)
    '経過秒数
    t = i * h

    '位置・速度
    x(1)  =  x(0) + h *    vx(0)
    y(1)  =  y(0) + h *    vy(0)
    vx(1) = vx(0) + h * fx(vx(0), vy(0))
    vy(1) = vy(0) + h * fy(vx(0), vy(0))

    x(2)  =  x(0) + h * (  vx(0)          +    vx(1)        ) / 2
    y(2)  =  y(0) + h * (  vy(0)          +    vy(1)        ) / 2
    vx(2) = vx(0) + h * (fx(vx(0), vy(0)) + fx(vx(1), vy(1))) / 2
    vy(2) = vy(0) + h * (fy(vx(0), vy(0)) + fy(vx(1), vy(1))) / 2

    x(0)  =  x(2)
    y(0)  =  y(2)
    vx(0) = vx(2)
    vy(0) = vy(2)

    WScript.StdOut.Write     Right(Space(4) & FormatNumber(t,     2, -1, 0, 0), 4) & vbTab
    WScript.StdOut.Write     Right(Space(8) & FormatNumber(vx(0), 5, -1, 0, 0), 8) & vbTab
    WScript.StdOut.Write     Right(Space(9) & FormatNumber(vy(0), 5, -1, 0, 0), 9) & vbTab
    WScript.StdOut.Write     Right(Space(9) & FormatNumber(x(0),  5, -1, 0, 0), 9) & vbTab
    WScript.StdOut.WriteLine Right(Space(8) & FormatNumber(y(0),  5, -1, 0, 0), 8)

    i = i + 1
Loop

Z:\>cscript //nologo Z:\0802.vbs
0.01    48.45620     48.35853     0.48622    0.48573
0.02    48.12690     47.93224     0.96912    0.96717
0.03    47.80238     47.51136     1.44876    1.44437
0.04    47.48254     47.09579     1.92517    1.91740
0.05    47.16726     46.68541     2.39841    2.38629
0.06    46.85647     46.28011     2.86852    2.85111
0.07    46.55007     45.87980     3.33554    3.31189
0.08    46.24796     45.48436     3.79952    3.76870
0.09    45.95006     45.09369     4.26050    4.22158
省略
6.20     9.25067    -23.74819   125.75002    2.41403
6.21     9.22711    -23.78559   125.84241    2.17636
6.22     9.20359    -23.82283   125.93456    1.93832
6.23     9.18010    -23.85991   126.02648    1.69991
6.24     9.15665    -23.89683   126.11816    1.46112
6.25     9.13323    -23.93360   126.20961    1.22197
6.26     9.10985    -23.97021   126.30083    0.98245
6.27     9.08651    -24.00667   126.39181    0.74256
6.28     9.06321    -24.04297   126.48256    0.50232
6.29     9.03994    -24.07911   126.57307    0.26171
6.30     9.01671    -24.11510   126.66335    0.02073
6.31     8.99351    -24.15094   126.75341   -0.22060

JScript

// 重力加速度
var g = -9.8
// 空気抵抗係数
var k = -0.01
// 時間間隔(秒)
var h = 0.01

// 角度
var degree = 45
var radian = degree * Math.PI / 180.0
// 初速 250 km/h -> 秒速に変換
var v = parseInt(250 * 1000 / 3600)
// 水平方向の速度
var vx = []
vx[0] = v * Math.cos(radian)
// 鉛直方向の速度
var vy = []
vy[0] = v * Math.sin(radian)
// 経過秒数
var t = 0.0
// 位置
var x = []
x[0] = 0.0
var y = []
y[0] = 0.0

// Heun法
for (var i = 1; y[0] >= 0.0; i++)
{
    // 経過秒数
    t = i * h

    // 位置・速度
    x[1]  =  x[0] + h *    vx[0]
    y[1]  =  y[0] + h *    vy[0]
    vx[1] = vx[0] + h * fx(vx[0], vy[0])
    vy[1] = vy[0] + h * fy(vx[0], vy[0])

    x[2]  =  x[0] + h * (  vx[0]          +    vx[1]        ) / 2
    y[2]  =  y[0] + h * (  vy[0]          +    vy[1]        ) / 2
    vx[2] = vx[0] + h * (fx(vx[0], vy[0]) + fx(vx[1], vy[1])) / 2
    vy[2] = vy[0] + h * (fy(vx[0], vy[0]) + fy(vx[1], vy[1])) / 2

    x[0]  =  x[2]
    y[0]  =  y[2]
    vx[0] = vx[2]
    vy[0] = vy[2]

    WScript.StdOut.Write(("    "      + t.toFixed(2)       ).slice(-4) + "\t")
    WScript.StdOut.Write(("        "  + vx[0].toFixed(5)   ).slice(-8) + "\t")
    WScript.StdOut.Write(("         " + vy[0].toFixed(5)   ).slice(-9) + "\t")
    WScript.StdOut.Write(("         " +  x[0].toFixed(5)   ).slice(-9) + "\t")
    WScript.StdOut.Write(("        "  +  y[0].toFixed(5)   ).slice(-8) + "\n")
}

// 空気抵抗による水平方向の減速分
function fx(vx, vy)
{
    return k * Math.sqrt(vx * vx + vy * vy) * vx
}
// 重力と空気抵抗による鉛直方向の減速分
function fy(vx, vy)
{
    return g + (k * Math.sqrt(vx * vx + vy * vy) * vy)
}

Z:\>cscript //nologo Z:\0802.js
0.01    48.45620     48.35853     0.48622    0.48573
0.02    48.12690     47.93224     0.96912    0.96717
0.03    47.80238     47.51136     1.44876    1.44437
0.04    47.48254     47.09579     1.92517    1.91740
0.05    47.16726     46.68541     2.39841    2.38629
0.06    46.85647     46.28011     2.86852    2.85111
0.07    46.55007     45.87980     3.33554    3.31189
0.08    46.24796     45.48436     3.79952    3.76870
0.09    45.95006     45.09369     4.26050    4.22158
省略
6.20     9.25067    -23.74819   125.75002    2.41403
6.21     9.22711    -23.78559   125.84241    2.17636
6.22     9.20359    -23.82283   125.93456    1.93832
6.23     9.18010    -23.85991   126.02648    1.69991
6.24     9.15665    -23.89683   126.11816    1.46112
6.25     9.13323    -23.93360   126.20961    1.22197
6.26     9.10985    -23.97021   126.30083    0.98245
6.27     9.08651    -24.00667   126.39181    0.74256
6.28     9.06321    -24.04297   126.48256    0.50232
6.29     9.03994    -24.07911   126.57307    0.26171
6.30     9.01671    -24.11510   126.66335    0.02073
6.31     8.99351    -24.15094   126.75341   -0.22060

PowerShell

# 重力加速度
$g = -9.8
# 空気抵抗係数
$k = -0.01
# 時間間隔(秒)
$h = 0.01

# 角度
$degree = 45
$radian = $degree * [Math]::PI / 180.0
# 初速 250 km/h -> 秒速に変換
$v = [Math]::Floor(250 * 1000 / 3600)
# 水平方向の速度
$vx  = New-Object double[] 3
$vx[0] = $v * [Math]::Cos($radian)
# 鉛直方向の速度
$vy  = New-Object double[] 3
$vy[0] = $v * [Math]::Sin($radian)
# 経過秒数
$t = 0.0
# 位置
$x  = New-Object double[] 3
$y  = New-Object double[] 3
$x[0] = 0.0
$y[0] = 0.0

# 空気抵抗による水平方向の減速分
function fx($vx, $vy)
{
    return $global:k * [Math]::Sqrt($vx * $vx + $vy * $vy) * $vx
}
# 重力と空気抵抗による鉛直方向の減速分
function fy($vx, $vy)
{
    return $global:g + ($global:k * [Math]::Sqrt($vx * $vx + $vy * $vy) * $vy)
}

# Heun法
for ($i = 1; $y[0] -ge 0.0; $i++)
{
    # 経過秒数
    $t = $i * $h

    # 位置・速度
    $x[1]  =  $x[0] + $h *     $vx[0]
    $y[1]  =  $y[0] + $h *     $vy[0]
    $vx[1] = $vx[0] + $h * (fx $vx[0] $vy[0])
    $vy[1] = $vy[0] + $h * (fy $vx[0] $vy[0])

    $x[2]  =  $x[0] + $h * (    $vx[0]         +     $vx[1]        ) / 2
    $y[2]  =  $y[0] + $h * (    $vy[0]         +     $vy[1]        ) / 2
    $vx[2] = $vx[0] + $h * ((fx $vx[0] $vy[0]) + (fx $vx[1] $vy[1])) / 2
    $vy[2] = $vy[0] + $h * ((fy $vx[0] $vy[0]) + (fy $vx[1] $vy[1])) / 2

    $x[0]  =  $x[2]
    $y[0]  =  $y[2]
    $vx[0] = $vx[2]
    $vy[0] = $vy[2]

    Write-Host ([String]::Format("{0,4:F2}`t{1,8:F5}`t{2,9:F5}`t{3,9:F5}`t{4,8:F5}", $t, $vx[0], $vy[0], $x[0], $y[0]))
}

Z:\>powershell -file Z:\0802.ps1
0.01    48.45620     48.35853     0.48622    0.48573
0.02    48.12690     47.93224     0.96912    0.96717
0.03    47.80238     47.51136     1.44876    1.44437
0.04    47.48254     47.09579     1.92517    1.91740
0.05    47.16726     46.68541     2.39841    2.38629
0.06    46.85647     46.28011     2.86852    2.85111
0.07    46.55007     45.87980     3.33554    3.31189
0.08    46.24796     45.48436     3.79952    3.76870
0.09    45.95006     45.09369     4.26050    4.22158
省略
6.20     9.25067    -23.74819   125.75002    2.41403
6.21     9.22711    -23.78559   125.84241    2.17636
6.22     9.20359    -23.82283   125.93456    1.93832
6.23     9.18010    -23.85991   126.02648    1.69991
6.24     9.15665    -23.89683   126.11816    1.46112
6.25     9.13323    -23.93360   126.20961    1.22197
6.26     9.10985    -23.97021   126.30083    0.98245
6.27     9.08651    -24.00667   126.39181    0.74256
6.28     9.06321    -24.04297   126.48256    0.50232
6.29     9.03994    -24.07911   126.57307    0.26171
6.30     9.01671    -24.11510   126.66335    0.02073
6.31     8.99351    -24.15094   126.75341   -0.22060

Perl

use Math::Trig 'pi';

# 重力加速度
my $g = -9.8;
# 空気抵抗係数
my $k = -0.01;
# 時間間隔(秒)
my $h = 0.01;

# 角度
my $degree = 45;
my $radian = $degree * pi / 180.0;
# 初速 250 km/h -> 秒速に変換
my $v = int(250 * 1000 / 3600);
# 水平方向の速度
my @vx = ();
$vx[0] = $v * cos($radian);
# 鉛直方向の速度
my @vy = ();
$vy[0] = $v * sin($radian);
# 経過秒数
my $t = 0.0;
# 位置
my @x = ();
$x[0] = 0.0;
my @y = ();
$y[0] = 0.0;

# Heun法
for (my $i = 1; $y[0] >= 0.0; $i++)
{
    # 経過秒数
    $t = $i * $h;

    # 位置・速度
    $x[1]  =  $x[0] + $h *    $vx[0];
    $y[1]  =  $y[0] + $h *    $vy[0];
    $vx[1] = $vx[0] + $h * fx($vx[0], $vy[0]);
    $vy[1] = $vy[0] + $h * fy($vx[0], $vy[0]);

    $x[2]  =  $x[0] + $h * (   $vx[0]          +    $vx[1]         ) / 2;
    $y[2]  =  $y[0] + $h * (   $vy[0]          +    $vy[1]         ) / 2;
    $vx[2] = $vx[0] + $h * (fx($vx[0], $vy[0]) + fx($vx[1], $vy[1])) / 2;
    $vy[2] = $vy[0] + $h * (fy($vx[0], $vy[0]) + fy($vx[1], $vy[1])) / 2;

    $x[0]  =  $x[2];
    $y[0]  =  $y[2];
    $vx[0] = $vx[2];
    $vy[0] = $vy[2];

    printf("%4.2f\t%8.5f\t%9.5f\t%9.5f\t%8.5f\n", $t, $vx[0], $vy[0], $x[0], $y[0]);
}

# 空気抵抗による水平方向の減速分
sub fx
{
  my ($vx, $vy) = @_;
    $k * sqrt($vx * $vx + $vy * $vy) * $vx;
}
# 重力と空気抵抗による鉛直方向の減速分
sub fy
{
    my ($vx, $vy) = @_;
    $g + ($k * sqrt($vx * $vx + $vy * $vy) * $vy);
}

Z:\>perl Z:\0802.pl
0.01    48.45620     48.35853     0.48622    0.48573
0.02    48.12690     47.93224     0.96912    0.96717
0.03    47.80238     47.51136     1.44876    1.44437
0.04    47.48254     47.09579     1.92517    1.91740
0.05    47.16726     46.68541     2.39841    2.38629
0.06    46.85647     46.28011     2.86852    2.85111
0.07    46.55007     45.87980     3.33554    3.31189
0.08    46.24796     45.48436     3.79952    3.76870
0.09    45.95006     45.09369     4.26050    4.22158
省略
6.20     9.25067    -23.74819   125.75002    2.41403
6.21     9.22711    -23.78559   125.84241    2.17636
6.22     9.20359    -23.82283   125.93456    1.93832
6.23     9.18010    -23.85991   126.02648    1.69991
6.24     9.15665    -23.89683   126.11816    1.46112
6.25     9.13323    -23.93360   126.20961    1.22197
6.26     9.10985    -23.97021   126.30083    0.98245
6.27     9.08651    -24.00667   126.39181    0.74256
6.28     9.06321    -24.04297   126.48256    0.50232
6.29     9.03994    -24.07911   126.57307    0.26171
6.30     9.01671    -24.11510   126.66335    0.02073
6.31     8.99351    -24.15094   126.75341   -0.22060

PHP

<?php

# 重力加速度
$g = -9.8;
# 空気抵抗係数
$k = -0.01;
# 時間間隔(秒)
$h = 0.01;

# 角度
$degree = 45;
$radian = $degree * M_PI / 180.0;
# 初速 250 km/h -> 秒速に変換
$v = (int)(250 * 1000 / 3600);
# 水平方向の速度
$vx = array();
$vx[0] = $v * cos($radian);
# 鉛直方向の速度
$vy = array();
$vy[0] = $v * sin($radian);
# 経過秒数
$t = 0.0;
# 位置
$x = array();
$x[0] = 0.0;
$y = array();
$y[0] = 0.0;

# Heun法
for ($i = 1; $y[0] >= 0.0; $i++)
{
    # 経過秒数
    $t = $i * $h;

    # 位置・速度
    $x[1]  =  $x[0] + $h *    $vx[0];
    $y[1]  =  $y[0] + $h *    $vy[0];
    $vx[1] = $vx[0] + $h * fx($vx[0], $vy[0]);
    $vy[1] = $vy[0] + $h * fy($vx[0], $vy[0]);

    $x[2]  =  $x[0] + $h * (   $vx[0]          +    $vx[1]         ) / 2;
    $y[2]  =  $y[0] + $h * (   $vy[0]          +    $vy[1]         ) / 2;
    $vx[2] = $vx[0] + $h * (fx($vx[0], $vy[0]) + fx($vx[1], $vy[1])) / 2;
    $vy[2] = $vy[0] + $h * (fy($vx[0], $vy[0]) + fy($vx[1], $vy[1])) / 2;

    $x[0]  =  $x[2];
    $y[0]  =  $y[2];
    $vx[0] = $vx[2];
    $vy[0] = $vy[2];

    printf("%4.2f\t%8.5f\t%9.5f\t%9.5f\t%8.5f\n", $t, $vx[0], $vy[0], $x[0], $y[0]);
}

# 空気抵抗による水平方向の減速分
function fx($vx, $vy)
{
    global $k;
    return $k * sqrt($vx * $vx + $vy * $vy) * $vx;
}
# 重力と空気抵抗による鉛直方向の減速分
function fy($vx, $vy)
{
    global $g, $k;
    return $g + ($k * sqrt($vx * $vx + $vy * $vy) * $vy);
}
?>

Z:\>php Z:\0802.php
0.01    48.45620     48.35853     0.48622    0.48573
0.02    48.12690     47.93224     0.96912    0.96717
0.03    47.80238     47.51136     1.44876    1.44437
0.04    47.48254     47.09579     1.92517    1.91740
0.05    47.16726     46.68541     2.39841    2.38629
0.06    46.85647     46.28011     2.86852    2.85111
0.07    46.55007     45.87980     3.33554    3.31189
0.08    46.24796     45.48436     3.79952    3.76870
0.09    45.95006     45.09369     4.26050    4.22158
省略
6.20     9.25067    -23.74819   125.75002    2.41403
6.21     9.22711    -23.78559   125.84241    2.17636
6.22     9.20359    -23.82283   125.93456    1.93832
6.23     9.18010    -23.85991   126.02648    1.69991
6.24     9.15665    -23.89683   126.11816    1.46112
6.25     9.13323    -23.93360   126.20961    1.22197
6.26     9.10985    -23.97021   126.30083    0.98245
6.27     9.08651    -24.00667   126.39181    0.74256
6.28     9.06321    -24.04297   126.48256    0.50232
6.29     9.03994    -24.07911   126.57307    0.26171
6.30     9.01671    -24.11510   126.66335    0.02073
6.31     8.99351    -24.15094   126.75341   -0.22060

Python

# coding: Shift_JIS

import math

# 重力加速度
g = -9.8
# 空気抵抗係数
k = -0.01
# 時間間隔(秒)
h = 0.01

# 角度
degree = 45
radian = degree * math.pi / 180.0
# 初速 250 km/h -> 秒速に変換
v = int(250 * 1000 / 3600)
# 水平方向の速度
vx = [0 for i in range(3)]
vx[0] = v * math.cos(radian)
# 鉛直方向の速度
vy = [0 for i in range(3)]
vy[0] = v * math.sin(radian)
# 経過秒数
t = 0.0
# 位置
x = [0 for i in range(3)]
x[0] = 0.0
y = [0 for i in range(3)]
y[0] = 0.0

# 空気抵抗による水平方向の減速分
def fx(vx, vy):
    return k * math.sqrt(vx * vx + vy * vy) * vx

# 重力と空気抵抗による鉛直方向の減速分
def fy(vx, vy):
    return g + (k * math.sqrt(vx * vx + vy * vy) * vy)

# Heun法
i = 1
while y[0] >= 0.0:
    # 経過秒数
    t = i * h

    # 位置・速度
    x[1]  =  x[0] + h *    vx[0]
    y[1]  =  y[0] + h *    vy[0]
    vx[1] = vx[0] + h * fx(vx[0], vy[0])
    vy[1] = vy[0] + h * fy(vx[0], vy[0])

    x[2]  =  x[0] + h * (  vx[0]          +    vx[1]        ) / 2
    y[2]  =  y[0] + h * (  vy[0]          +    vy[1]        ) / 2
    vx[2] = vx[0] + h * (fx(vx[0], vy[0]) + fx(vx[1], vy[1])) / 2
    vy[2] = vy[0] + h * (fy(vx[0], vy[0]) + fy(vx[1], vy[1])) / 2

    x[0]  =  x[2]
    y[0]  =  y[2]
    vx[0] = vx[2]
    vy[0] = vy[2]

    print "%4.2f\t%8.5f\t%9.5f\t%9.5f\t%8.5f" % (t, vx[0], vy[0], x[0], y[0])

    i += 1

Z:\>python Z:\0802.py
0.01    48.45620     48.35853     0.48622    0.48573
0.02    48.12690     47.93224     0.96912    0.96717
0.03    47.80238     47.51136     1.44876    1.44437
0.04    47.48254     47.09579     1.92517    1.91740
0.05    47.16726     46.68541     2.39841    2.38629
0.06    46.85647     46.28011     2.86852    2.85111
0.07    46.55007     45.87980     3.33554    3.31189
0.08    46.24796     45.48436     3.79952    3.76870
0.09    45.95006     45.09369     4.26050    4.22158
省略
6.20     9.25067    -23.74819   125.75002    2.41403
6.21     9.22711    -23.78559   125.84241    2.17636
6.22     9.20359    -23.82283   125.93456    1.93832
6.23     9.18010    -23.85991   126.02648    1.69991
6.24     9.15665    -23.89683   126.11816    1.46112
6.25     9.13323    -23.93360   126.20961    1.22197
6.26     9.10985    -23.97021   126.30083    0.98245
6.27     9.08651    -24.00667   126.39181    0.74256
6.28     9.06321    -24.04297   126.48256    0.50232
6.29     9.03994    -24.07911   126.57307    0.26171
6.30     9.01671    -24.11510   126.66335    0.02073
6.31     8.99351    -24.15094   126.75341   -0.22060

Ruby

# 重力加速度
$g = -9.8
# 空気抵抗係数
$k = -0.01
# 時間間隔(秒)
h = 0.01

# 角度
degree = 45
radian = degree * Math::PI / 180.0
# 初速 250 km/h -> 秒速に変換
v = 250 * 1000 / 3600
# 水平方向の速度
vx = Array.new(3)
vx[0] = v * Math.cos(radian)
# 鉛直方向の速度
vy = Array.new(3)
vy[0] = v * Math.sin(radian)
# 経過秒数
t = 0.0
# 位置
x = Array.new(3)
x[0] = 0.0
y = Array.new(3)
y[0] = 0.0

# 空気抵抗による水平方向の減速分
def fx(vx, vy)
    return $k * Math.sqrt(vx * vx + vy * vy) * vx
end

# 重力と空気抵抗による鉛直方向の減速分
def fy(vx, vy)
    return $g + ($k * Math.sqrt(vx * vx + vy * vy) * vy)
end

# Heun法
i = 1
while y[0] >= 0.0 do
    # 経過秒数
    t = i * h

    # 位置・速度
    x[1]  =  x[0] + h *    vx[0]
    y[1]  =  y[0] + h *    vy[0]
    vx[1] = vx[0] + h * fx(vx[0], vy[0])
    vy[1] = vy[0] + h * fy(vx[0], vy[0])

    x[2]  =  x[0] + h * (  vx[0]          +    vx[1]        ) / 2
    y[2]  =  y[0] + h * (  vy[0]          +    vy[1]        ) / 2
    vx[2] = vx[0] + h * (fx(vx[0], vy[0]) + fx(vx[1], vy[1])) / 2
    vy[2] = vy[0] + h * (fy(vx[0], vy[0]) + fy(vx[1], vy[1])) / 2

    x[0]  =  x[2]
    y[0]  =  y[2]
    vx[0] = vx[2]
    vy[0] = vy[2]

    printf("%4.2f\t%8.5f\t%9.5f\t%9.5f\t%8.5f\n", t, vx[0], vy[0], x[0], y[0])

    i += 1
end

Z:\>ruby Z:\0802.rb
0.01    48.45620     48.35853     0.48622    0.48573
0.02    48.12690     47.93224     0.96912    0.96717
0.03    47.80238     47.51136     1.44876    1.44437
0.04    47.48254     47.09579     1.92517    1.91740
0.05    47.16726     46.68541     2.39841    2.38629
0.06    46.85647     46.28011     2.86852    2.85111
0.07    46.55007     45.87980     3.33554    3.31189
0.08    46.24796     45.48436     3.79952    3.76870
0.09    45.95006     45.09369     4.26050    4.22158
省略
6.20     9.25067    -23.74819   125.75002    2.41403
6.21     9.22711    -23.78559   125.84241    2.17636
6.22     9.20359    -23.82283   125.93456    1.93832
6.23     9.18010    -23.85991   126.02648    1.69991
6.24     9.15665    -23.89683   126.11816    1.46112
6.25     9.13323    -23.93360   126.20961    1.22197
6.26     9.10985    -23.97021   126.30083    0.98245
6.27     9.08651    -24.00667   126.39181    0.74256
6.28     9.06321    -24.04297   126.48256    0.50232
6.29     9.03994    -24.07911   126.57307    0.26171
6.30     9.01671    -24.11510   126.66335    0.02073
6.31     8.99351    -24.15094   126.75341   -0.22060

Groovy

Pascal

program Pas0802(arg);
{$MODE delphi}

uses
    SysUtils, Math;

const
    // 重力加速度
    g = -9.8;
    // 空気抵抗係数
    k = -0.01;
    // 時間間隔(秒)
    h = 0.01;
    // 角度
    degree = 45;

// 空気抵抗による水平方向の減速分
function fx(vx:Double; vy:Double):Double;
begin
    result := k * Sqrt(vx * vx + vy * vy) * vx;
end;

// 重力と空気抵抗による鉛直方向の減速分
function fy(vx:Double; vy:Double):Double;
begin
    result := g + (k * Sqrt(vx * vx + vy * vy) * vy);
end;

var
    // 角度
    radian:Double;
    // 初速
    v:Double;
    // 水平方向の速度
    vx:array [0..2] of Double;
    // 鉛直方向の速度
    vy:array [0..2] of Double;
    // 経過秒数
    t:Double = 0.0;
    // 位置
    x:array [0..2] of Double;
    y:array [0..2] of Double;

    i:Integer;
begin
    // 角度
    radian := degree * PI / 180.0;
    // 初速 250 km/h -> 秒速に変換
    v := 250 * 1000 div 3600;
    // 水平方向の速度
    vx[0] := v * Cos(radian);
    // 鉛直方向の速度
    vy[0] := v * Sin(radian);
    // 位置
    x[0] := 0.0;
    y[0] := 0.0;

    // Heun法
    i := 1;
    while y[0] >= 0.0 do
    begin
        // 経過秒数
        t := i * h;

        // 位置・速度
        x[1]  :=  x[0] + h *    vx[0];
        y[1]  :=  y[0] + h *    vy[0];
        vx[1] := vx[0] + h * fx(vx[0], vy[0]);
        vy[1] := vy[0] + h * fy(vx[0], vy[0]);

        x[2]  :=  x[0] + h * (  vx[0]          +    vx[1]        ) / 2;
        y[2]  :=  y[0] + h * (  vy[0]          +    vy[1]        ) / 2;
        vx[2] := vx[0] + h * (fx(vx[0], vy[0]) + fx(vx[1], vy[1])) / 2;
        vy[2] := vy[0] + h * (fy(vx[0], vy[0]) + fy(vx[1], vy[1])) / 2;

        x[0]  :=  x[2];
        y[0]  :=  y[2];
        vx[0] := vx[2];
        vy[0] := vy[2];

        writeln(format('%4.2f'#9'%8.5f'#9'%9.5f'#9'%9.5f'#9'%8.5f', [t, vx[0], vy[0], x[0], y[0]]));

        inc(i);
    end;
end.

Z:\>fpc -v0 -l- Pas0802.pp

Z:\>Pas0802
0.01    48.45620     48.35853     0.48622    0.48573
0.02    48.12690     47.93224     0.96912    0.96717
0.03    47.80238     47.51136     1.44876    1.44437
0.04    47.48254     47.09579     1.92517    1.91740
0.05    47.16726     46.68541     2.39841    2.38629
0.06    46.85647     46.28011     2.86852    2.85111
0.07    46.55007     45.87980     3.33554    3.31189
0.08    46.24796     45.48436     3.79952    3.76870
0.09    45.95006     45.09369     4.26050    4.22158
省略
6.20     9.25067    -23.74819   125.75002    2.41403
6.21     9.22711    -23.78559   125.84241    2.17636
6.22     9.20359    -23.82283   125.93456    1.93832
6.23     9.18010    -23.85991   126.02648    1.69991
6.24     9.15665    -23.89683   126.11816    1.46112
6.25     9.13323    -23.93360   126.20961    1.22197
6.26     9.10985    -23.97021   126.30083    0.98245
6.27     9.08651    -24.00667   126.39181    0.74256
6.28     9.06321    -24.04297   126.48256    0.50232
6.29     9.03994    -24.07911   126.57307    0.26171
6.30     9.01671    -24.11510   126.66335    0.02073
6.31     8.99351    -24.15094   126.75341   -0.22060

Ada

with TEXT_IO, Ada.Long_Float_Text_IO, Ada.Numerics, Ada.Numerics.Long_Elementary_Functions;
use  TEXT_IO, Ada.Long_Float_Text_IO, Ada.Numerics, Ada.Numerics.Long_Elementary_Functions;

procedure Ada0802 is
    -- 重力加速度
    g : Constant Long_Float := -9.8;
    -- 空気抵抗係数
    k : Constant Long_Float := -0.01;
    -- 時間間隔(秒)
    h : Constant Long_Float := 0.01;
    -- 角度
    degree : Constant Long_Float := 45.0;

    -- 空気抵抗による水平方向の減速分
    function fx(vx:Long_Float; vy:Long_Float) return Long_Float is
    begin
        return k * Sqrt(vx * vx + vy * vy) * vx;
    end fx;

    -- 重力と空気抵抗による鉛直方向の減速分
    function fy(vx:Long_Float; vy:Long_Float) return Long_Float is
    begin
        return g + (k * Sqrt(vx * vx + vy * vy) * vy);
    end fy;

    -- 角度
    radian:Long_Float;
    -- 初速
    v:Long_Float;
    -- 水平方向の速度
    vx:array (0..2) of Long_Float;
    -- 鉛直方向の速度
    vy:array (0..2) of Long_Float;
    -- 経過秒数
    t:Long_Float := 0.0;
    -- 位置
    x:array (0..2) of Long_Float;
    y:array (0..2) of Long_Float;

    i:Integer;
begin
    -- 角度
    radian := degree * Pi / 180.0;
    -- 初速 250 km/h -> 秒速に変換
    v := Long_Float(250 * 1000 / 3600);
    -- 水平方向の速度
    vx(0) := v * Cos(radian);
    -- 鉛直方向の速度
    vy(0) := v * Sin(radian);
    -- 位置
    x(0) := 0.0;
    y(0) := 0.0;

    -- Heun法
    i := 1;
    while y(0) >= 0.0 loop
        -- 経過秒数
        t := Long_Float(i) * h;

        -- 位置・速度
        x(1)  :=  x(0) + h *    vx(0);
        y(1)  :=  y(0) + h *    vy(0);
        vx(1) := vx(0) + h * fx(vx(0), vy(0));
        vy(1) := vy(0) + h * fy(vx(0), vy(0));

        x(2)  :=  x(0) + h * (  vx(0)          +    vx(1)        ) / 2.0;
        y(2)  :=  y(0) + h * (  vy(0)          +    vy(1)        ) / 2.0;
        vx(2) := vx(0) + h * (fx(vx(0), vy(0)) + fx(vx(1), vy(1))) / 2.0;
        vy(2) := vy(0) + h * (fy(vx(0), vy(0)) + fy(vx(1), vy(1))) / 2.0;

        x(0)  :=  x(2);
        y(0)  :=  y(2);
        vx(0) := vx(2);
        vy(0) := vy(2);

        Put(t,       Fore=>1, Aft=>2, Exp=>0);
        Put(Ascii.HT);
        Put(vx(0),   Fore=>3, Aft=>5, Exp=>0);
        Put(Ascii.HT);
        Put(vy(0),   Fore=>4, Aft=>5, Exp=>0);
        Put(Ascii.HT);
        Put(x(0),    Fore=>4, Aft=>5, Exp=>0);
        Put(Ascii.HT);
        Put(y(0),    Fore=>4, Aft=>5, Exp=>0);
        New_Line;

        i := i + 1;
    end loop;
end Ada0802;

xxxxxx@yyyyyy /Z
$ gnatmake Ada0802.adb

xxxxxx@yyyyyy /Z
$ Ada0802
0.01     48.45620     48.35853     0.48622     0.48573
0.02     48.12690     47.93224     0.96912     0.96717
0.03     47.80238     47.51136     1.44876     1.44437
0.04     47.48254     47.09579     1.92517     1.91740
0.05     47.16726     46.68541     2.39841     2.38629
0.06     46.85647     46.28011     2.86852     2.85111
0.07     46.55007     45.87980     3.33554     3.31189
0.08     46.24796     45.48436     3.79952     3.76870
0.09     45.95006     45.09369     4.26050     4.22158
省略
6.20      9.25067    -23.74819   125.75002     2.41403
6.21      9.22711    -23.78559   125.84241     2.17636
6.22      9.20359    -23.82283   125.93456     1.93832
6.23      9.18010    -23.85991   126.02648     1.69991
6.24      9.15665    -23.89683   126.11816     1.46112
6.25      9.13323    -23.93360   126.20961     1.22197
6.26      9.10985    -23.97021   126.30083     0.98245
6.27      9.08651    -24.00667   126.39181     0.74256
6.28      9.06321    -24.04297   126.48256     0.50232
6.29      9.03994    -24.07911   126.57307     0.26171
6.30      9.01671    -24.11510   126.66335     0.02073
6.31      8.99351    -24.15094   126.75341    -0.22060

VB.NET

Option Explicit

Module VB0802
    '重力加速度
    Private Const g As Double = -9.8
    '空気抵抗係数
    Private Const k As Double = -0.01
    '時間間隔(秒)
    Private Const h As Double = 0.01

    Public Sub Main()
        '角度
        Const degree As Double = 45
        Dim radian As Double = degree * Math.PI / 180.0
        '初速 250 km/h -> 秒速に変換
        Dim v As Double = 250 * 1000 \ 3600
        '水平方向の速度
        Dim vx(2) As Double
        vx(0) = v * Math.Cos(radian)
        '鉛直方向の速度
        Dim vy(2) As Double
        vy(0) = v * Math.Sin(radian)
        '経過秒数
        Dim t As Double = 0.0
        '位置
        Dim x(2) As Double
        x(0) = 0.0
        Dim y(2) As Double
        y(0) = 0.0

        'Heun法
        Dim i As Integer = 1
        Do While (y(0) >= 0.0)
            '経過秒数
            t = i * h

            '位置・速度
            x(1)  =  x(0) + h *    vx(0)
            y(1)  =  y(0) + h *    vy(0)
            vx(1) = vx(0) + h * fx(vx(0), vy(0))
            vy(1) = vy(0) + h * fy(vx(0), vy(0))

            x(2)  =  x(0) + h * (  vx(0)          +    vx(1)        ) / 2
            y(2)  =  y(0) + h * (  vy(0)          +    vy(1)        ) / 2
            vx(2) = vx(0) + h * (fx(vx(0), vy(0)) + fx(vx(1), vy(1))) / 2
            vy(2) = vy(0) + h * (fy(vx(0), vy(0)) + fy(vx(1), vy(1))) / 2

            x(0)  =  x(2)
            y(0)  =  y(2)
            vx(0) = vx(2)
            vy(0) = vy(2)

            Console.WriteLine(String.Format("{0,4:F2}{5}{1,8:F5}{5}{2,9:F5}{5}{3,9:F5}{5}{4,8:F5}", t, vx(0), vy(0), x(0), y(0), vbTab))

            i += 1
        Loop
    End Sub

    '空気抵抗による水平方向の減速分
    Private Function fx(ByVal vx As Double, ByVal vy As Double) As Double
        Return k * Math.Sqrt(vx * vx + vy * vy) * vx
    End Function

    '重力と空気抵抗による鉛直方向の減速分
    Private Function fy(ByVal vx As Double, ByVal vy As Double) As Double
        Return g + (k * Math.Sqrt(vx * vx + vy * vy) * vy)
    End Function
End Module

Z:\>vbc -nologo VB0802.vb

Z:\>VB0802
0.01    48.45620     48.35853     0.48622    0.48573
0.02    48.12690     47.93224     0.96912    0.96717
0.03    47.80238     47.51136     1.44876    1.44437
0.04    47.48254     47.09579     1.92517    1.91740
0.05    47.16726     46.68541     2.39841    2.38629
0.06    46.85647     46.28011     2.86852    2.85111
0.07    46.55007     45.87980     3.33554    3.31189
0.08    46.24796     45.48436     3.79952    3.76870
0.09    45.95006     45.09369     4.26050    4.22158
省略
6.20     9.25067    -23.74819   125.75002    2.41403
6.21     9.22711    -23.78559   125.84241    2.17636
6.22     9.20359    -23.82283   125.93456    1.93832
6.23     9.18010    -23.85991   126.02648    1.69991
6.24     9.15665    -23.89683   126.11816    1.46112
6.25     9.13323    -23.93360   126.20961    1.22197
6.26     9.10985    -23.97021   126.30083    0.98245
6.27     9.08651    -24.00667   126.39181    0.74256
6.28     9.06321    -24.04297   126.48256    0.50232
6.29     9.03994    -24.07911   126.57307    0.26171
6.30     9.01671    -24.11510   126.66335    0.02073
6.31     8.99351    -24.15094   126.75341   -0.22060

C#

using System;

public class CS0802
{
    // 重力加速度
    private const double g = -9.8;
    // 空気抵抗係数
    private const double k = -0.01;
    // 時間間隔(秒)
    private const double h = 0.01;

    public static void Main()
    {
        // 角度
        double degree = 45;
        double radian = degree * Math.PI / 180.0;
        // 初速 250 km/h -> 秒速に変換
        double v = 250 * 1000 / 3600;
        // 水平方向の速度
        double[] vx = new double[3];
        vx[0] = v * Math.Cos(radian);
        // 鉛直方向の速度
        double[] vy = new double[3];
        vy[0] = v * Math.Sin(radian);
        // 経過秒数
        double t = 0.0;
        // 位置
        double[] x = new double[3];
        x[0] = 0.0;
        double[] y = new double[3];
        y[0] = 0.0;

        // Heun法
        for (int i = 1; y[0] >= 0.0; i++)
        {
            // 経過秒数
            t = i * h;

            // 位置・速度
            x[1]  =  x[0] + h *    vx[0];
            y[1]  =  y[0] + h *    vy[0];
            vx[1] = vx[0] + h * fx(vx[0], vy[0]);
            vy[1] = vy[0] + h * fy(vx[0], vy[0]);

            x[2]  =  x[0] + h * (  vx[0]          +    vx[1]        ) / 2;
            y[2]  =  y[0] + h * (  vy[0]          +    vy[1]        ) / 2;
            vx[2] = vx[0] + h * (fx(vx[0], vy[0]) + fx(vx[1], vy[1])) / 2;
            vy[2] = vy[0] + h * (fy(vx[0], vy[0]) + fy(vx[1], vy[1])) / 2;

            x[0]  =  x[2];
            y[0]  =  y[2];
            vx[0] = vx[2];
            vy[0] = vy[2];

            Console.WriteLine(string.Format("{0,4:F2}\t{1,8:F5}\t{2,9:F5}\t{3,9:F5}\t{4,8:F5}", t, vx[0], vy[0], x[0], y[0]));
        }
    }

    // 空気抵抗による水平方向の減速分
    private static double fx(double vx, double vy)
    {
        return k * Math.Sqrt(vx * vx + vy * vy) * vx;
    }
    // 重力と空気抵抗による鉛直方向の減速分
    private static double fy(double vx, double vy)
    {
        return g + (k * Math.Sqrt(vx * vx + vy * vy) * vy);
    }
}

Z:\>csc -nologo CS0802.cs

Z:\>CS0802
0.01    48.45620     48.35853     0.48622    0.48573
0.02    48.12690     47.93224     0.96912    0.96717
0.03    47.80238     47.51136     1.44876    1.44437
0.04    47.48254     47.09579     1.92517    1.91740
0.05    47.16726     46.68541     2.39841    2.38629
0.06    46.85647     46.28011     2.86852    2.85111
0.07    46.55007     45.87980     3.33554    3.31189
0.08    46.24796     45.48436     3.79952    3.76870
0.09    45.95006     45.09369     4.26050    4.22158
省略
6.20     9.25067    -23.74819   125.75002    2.41403
6.21     9.22711    -23.78559   125.84241    2.17636
6.22     9.20359    -23.82283   125.93456    1.93832
6.23     9.18010    -23.85991   126.02648    1.69991
6.24     9.15665    -23.89683   126.11816    1.46112
6.25     9.13323    -23.93360   126.20961    1.22197
6.26     9.10985    -23.97021   126.30083    0.98245
6.27     9.08651    -24.00667   126.39181    0.74256
6.28     9.06321    -24.04297   126.48256    0.50232
6.29     9.03994    -24.07911   126.57307    0.26171
6.30     9.01671    -24.11510   126.66335    0.02073
6.31     8.99351    -24.15094   126.75341   -0.22060

Java

import static java.lang.System.out;

public class Java0802 {

    // 重力加速度
    private static final double g = -9.8;
    // 空気抵抗係数
    private static final double k = -0.01;
    // 時間間隔(秒)
    private static final double h = 0.01;

    public static void main(String []args) {
        // 角度
        double degree = 45;
        double radian = degree * Math.PI / 180.0;
        // 初速 250 km/h -> 秒速に変換
        double v = 250 * 1000 / 3600;
        // 水平方向の速度
        double[] vx = new double[3];
        vx[0] = v * Math.cos(radian);
        // 鉛直方向の速度
        double[] vy = new double[3];
        vy[0] = v * Math.sin(radian);
        // 経過秒数
        double t = 0.0;
        // 位置
        double[] x = new double[3];
        x[0] = 0.0;
        double[] y = new double[3];
        y[0] = 0.0;

        // Heun法
        for (int i = 1; y[0] >= 0.0; i++) {
            // 経過秒数
            t = i * h;

            // 位置・速度
            x[1]  =  x[0] + h *    vx[0];
            y[1]  =  y[0] + h *    vy[0];
            vx[1] = vx[0] + h * fx(vx[0], vy[0]);
            vy[1] = vy[0] + h * fy(vx[0], vy[0]);

            x[2]  =  x[0] + h * (  vx[0]          +    vx[1]        ) / 2;
            y[2]  =  y[0] + h * (  vy[0]          +    vy[1]        ) / 2;
            vx[2] = vx[0] + h * (fx(vx[0], vy[0]) + fx(vx[1], vy[1])) / 2;
            vy[2] = vy[0] + h * (fy(vx[0], vy[0]) + fy(vx[1], vy[1])) / 2;

            x[0]  =  x[2];
            y[0]  =  y[2];
            vx[0] = vx[2];
            vy[0] = vy[2];

            out.println(String.format("%4.2f\t%8.5f\t%9.5f\t%9.5f\t%8.5f", t, vx[0], vy[0], x[0], y[0]));
        }
    }

    // 空気抵抗による水平方向の減速分
    private static double fx(double vx, double vy) {
        return k * Math.sqrt(vx * vx + vy * vy) * vx;
    }
    // 重力と空気抵抗による鉛直方向の減速分
    private static double fy(double vx, double vy) {
        return g + (k * Math.sqrt(vx * vx + vy * vy) * vy);
    }
}

Z:\>javac Java0802.java

Z:\>java Java0802
0.01    48.45620     48.35853     0.48622    0.48573
0.02    48.12690     47.93224     0.96912    0.96717
0.03    47.80238     47.51136     1.44876    1.44437
0.04    47.48254     47.09579     1.92517    1.91740
0.05    47.16726     46.68541     2.39841    2.38629
0.06    46.85647     46.28011     2.86852    2.85111
0.07    46.55007     45.87980     3.33554    3.31189
0.08    46.24796     45.48436     3.79952    3.76870
0.09    45.95006     45.09369     4.26050    4.22158
省略
6.20     9.25067    -23.74819   125.75002    2.41403
6.21     9.22711    -23.78559   125.84241    2.17636
6.22     9.20359    -23.82283   125.93456    1.93832
6.23     9.18010    -23.85991   126.02648    1.69991
6.24     9.15665    -23.89683   126.11816    1.46112
6.25     9.13323    -23.93360   126.20961    1.22197
6.26     9.10985    -23.97021   126.30083    0.98245
6.27     9.08651    -24.00667   126.39181    0.74256
6.28     9.06321    -24.04297   126.48256    0.50232
6.29     9.03994    -24.07911   126.57307    0.26171
6.30     9.01671    -24.11510   126.66335    0.02073
6.31     8.99351    -24.15094   126.75341   -0.22060

C++

#include <iostream>
#include <iomanip>
#include <math.h>
using namespace std;

// 重力加速度
const double g = -9.8;
// 空気抵抗係数
const double k = -0.01;
// 時間間隔(秒)
const double h = 0.01;

// 空気抵抗による水平方向の減速分
double fx(double vx, double vy);
// 重力と空気抵抗による鉛直方向の減速分
double fy(double vx, double vy);

int main()
{
    // 角度
    double degree = 45;
    double radian = degree * M_PI / 180.0;
    // 初速 250 km/h -> 秒速に変換
    double v = 250 * 1000 / 3600;
    // 水平方向の速度
    double vx[3];
    vx[0] = v * cos(radian);
    // 鉛直方向の速度
    double vy[3];
    vy[0] = v * sin(radian);
    // 経過秒数
    double t = 0.0;
    // 位置
    double x[3];
    x[0] = 0.0;
    double y[3];
    y[0] = 0.0;

    // Heun法
    for (int i = 1; y[0] >= 0.0; i++)
    {
        // 経過秒数
        t = i * h;
        cout << setw(4) << fixed << setprecision(2) << t << "\t";

        // 位置・速度
        x[1]  =  x[0] + h *    vx[0];
        y[1]  =  y[0] + h *    vy[0];
        vx[1] = vx[0] + h * fx(vx[0], vy[0]);
        vy[1] = vy[0] + h * fy(vx[0], vy[0]);

        x[2]  =  x[0] + h * (  vx[0]          +    vx[1]        ) / 2;
        y[2]  =  y[0] + h * (  vy[0]          +    vy[1]        ) / 2;
        vx[2] = vx[0] + h * (fx(vx[0], vy[0]) + fx(vx[1], vy[1])) / 2;
        vy[2] = vy[0] + h * (fy(vx[0], vy[0]) + fy(vx[1], vy[1])) / 2;

        x[0]  =  x[2];
        y[0]  =  y[2];
        vx[0] = vx[2];
        vy[0] = vy[2];

        cout << setw(8) << fixed << setprecision(5) << vx[0] << "\t";
        cout << setw(9) << fixed << setprecision(5) << vy[0] << "\t";
        cout << setw(9) << fixed << setprecision(5) <<  x[0] << "\t";
        cout << setw(8) << fixed << setprecision(5) <<  y[0] << endl;
    }
    return 0;
}

// 空気抵抗による水平方向の減速分
double fx(double vx, double vy)
{
    return k * sqrt(vx * vx + vy * vy) * vx;
}
// 重力と空気抵抗による鉛直方向の減速分
double fy(double vx, double vy)
{
    return g + (k * sqrt(vx * vx + vy * vy) * vy);
}

Z:\>bcc32 -q CP0802.cpp
cp0802.cpp:

Z:\>CP0802
0.01    48.45620     48.35853     0.48622    0.48573
0.02    48.12690     47.93224     0.96912    0.96717
0.03    47.80238     47.51136     1.44876    1.44437
0.04    47.48254     47.09579     1.92517    1.91740
0.05    47.16726     46.68541     2.39841    2.38629
0.06    46.85647     46.28011     2.86852    2.85111
0.07    46.55007     45.87980     3.33554    3.31189
0.08    46.24796     45.48436     3.79952    3.76870
0.09    45.95006     45.09369     4.26050    4.22158
省略
6.20     9.25067    -23.74819   125.75002    2.41403
6.21     9.22711    -23.78559   125.84241    2.17636
6.22     9.20359    -23.82283   125.93456    1.93832
6.23     9.18010    -23.85991   126.02648    1.69991
6.24     9.15665    -23.89683   126.11816    1.46112
6.25     9.13323    -23.93360   126.20961    1.22197
6.26     9.10985    -23.97021   126.30083    0.98245
6.27     9.08651    -24.00667   126.39181    0.74256
6.28     9.06321    -24.04297   126.48256    0.50232
6.29     9.03994    -24.07911   126.57307    0.26171
6.30     9.01671    -24.11510   126.66335    0.02073
6.31     8.99351    -24.15094   126.75341   -0.22060

Objective-C

#import <Foundation/Foundation.h>
#import <math.h>

// 重力加速度
const double g = -9.8;
// 空気抵抗係数
const double k = -0.01;
// 時間間隔(秒)
const double h = 0.01;

// 空気抵抗による水平方向の減速分
double fx(double vx, double vy);
// 重力と空気抵抗による鉛直方向の減速分
double fy(double vx, double vy);

int main()
{
    // 角度
    double degree = 45;
    double radian = degree * M_PI / 180.0;
    // 初速 250 km/h -> 秒速に変換
    double v = 250 * 1000 / 3600;
    // 水平方向の速度
    double vx[3];
    vx[0] = v * cos(radian);
    // 鉛直方向の速度
    double vy[3];
    vy[0] = v * sin(radian);
    // 経過秒数
    double t = 0.0;
    // 位置
    double x[3];
    x[0] = 0.0;
    double y[3];
    y[0] = 0.0;

    // Heun法
    int i;
    for (i = 1; y[0] >= 0.0; i++)
    {
        // 経過秒数
        t = i * h;

        // 位置・速度
        x[1]  =  x[0] + h *    vx[0];
        y[1]  =  y[0] + h *    vy[0];
        vx[1] = vx[0] + h * fx(vx[0], vy[0]);
        vy[1] = vy[0] + h * fy(vx[0], vy[0]);

        x[2]  =  x[0] + h * (  vx[0]          +    vx[1]        ) / 2;
        y[2]  =  y[0] + h * (  vy[0]          +    vy[1]        ) / 2;
        vx[2] = vx[0] + h * (fx(vx[0], vy[0]) + fx(vx[1], vy[1])) / 2;
        vy[2] = vy[0] + h * (fy(vx[0], vy[0]) + fy(vx[1], vy[1])) / 2;

        x[0]  =  x[2];
        y[0]  =  y[2];
        vx[0] = vx[2];
        vy[0] = vy[2];

        printf("%4.2f\t%8.5f\t%9.5f\t%9.5f\t%8.5f\n", t, vx[0], vy[0], x[0], y[0]);
    }
    return 0;
}

// 空気抵抗による水平方向の減速分
double fx(double vx, double vy)
{
    return k * sqrt(vx * vx + vy * vy) * vx;
}
// 重力と空気抵抗による鉛直方向の減速分
double fy(double vx, double vy)
{
    return g + (k * sqrt(vx * vx + vy * vy) * vy);
}

xxxxxx@yyyyyy /Z
$ gcc -o OC0802 OC0802.m -lobjc -lgnustep-base -I $INCLUDE -L $LIB $CFLAGS

xxxxxx@yyyyyy /Z
$ OC0802
0.01    48.45620     48.35853     0.48622    0.48573
0.02    48.12690     47.93224     0.96912    0.96717
0.03    47.80238     47.51136     1.44876    1.44437
0.04    47.48254     47.09579     1.92517    1.91740
0.05    47.16726     46.68541     2.39841    2.38629
0.06    46.85647     46.28011     2.86852    2.85111
0.07    46.55007     45.87980     3.33554    3.31189
0.08    46.24796     45.48436     3.79952    3.76870
0.09    45.95006     45.09369     4.26050    4.22158
省略
6.20     9.25067    -23.74819   125.75002    2.41403
6.21     9.22711    -23.78559   125.84241    2.17636
6.22     9.20359    -23.82283   125.93456    1.93832
6.23     9.18010    -23.85991   126.02648    1.69991
6.24     9.15665    -23.89683   126.11816    1.46112
6.25     9.13323    -23.93360   126.20961    1.22197
6.26     9.10985    -23.97021   126.30083    0.98245
6.27     9.08651    -24.00667   126.39181    0.74256
6.28     9.06321    -24.04297   126.48256    0.50232
6.29     9.03994    -24.07911   126.57307    0.26171
6.30     9.01671    -24.11510   126.66335    0.02073
6.31     8.99351    -24.15094   126.75341   -0.22060

D

import std.stdio;
import std.math;

// 重力加速度
const double g = -9.8;
// 空気抵抗係数
const double k = -0.01;
// 時間間隔(秒)
const double h = 0.01;

void main(string[] args)
{
    // 角度
    double degree = 45;
    double radian = degree * PI / 180.0;
    // 初速 250 km/h -> 秒速に変換
    double v = 250 * 1000 / 3600;
    // 水平方向の速度
    double vx[3];
    vx[0] = v * cos(radian);
    // 鉛直方向の速度
    double vy[3];
    vy[0] = v * sin(radian);
    // 経過秒数
    double t = 0.0;
    // 位置
    double x[3];
    x[0] = 0.0;
    double y[3];
    y[0] = 0.0;

    // Heun法
    for (int i = 1; y[0] >= 0.0; i++)
    {
        // 経過秒数
        t = i * h;

        // 位置・速度
        x[1]  =  x[0] + h *    vx[0];
        y[1]  =  y[0] + h *    vy[0];
        vx[1] = vx[0] + h * fx(vx[0], vy[0]);
        vy[1] = vy[0] + h * fy(vx[0], vy[0]);

        x[2]  =  x[0] + h * (  vx[0]          +    vx[1]        ) / 2;
        y[2]  =  y[0] + h * (  vy[0]          +    vy[1]        ) / 2;
        vx[2] = vx[0] + h * (fx(vx[0], vy[0]) + fx(vx[1], vy[1])) / 2;
        vy[2] = vy[0] + h * (fy(vx[0], vy[0]) + fy(vx[1], vy[1])) / 2;

        x[0]  =  x[2];
        y[0]  =  y[2];
        vx[0] = vx[2];
        vy[0] = vy[2];

        writefln("%4.2f\t%8.5f\t%9.5f\t%9.5f\t%9.5f", t, vx[0], vy[0], x[0], y[0]);
    }
}

// 空気抵抗による水平方向の減速分
double fx(double vx, double vy)
{
    return k * sqrt(vx * vx + vy * vy) * vx;
}
// 重力と空気抵抗による鉛直方向の減速分
double fy(double vx, double vy)
{
    return g + (k * sqrt(vx * vx + vy * vy) * vy);
}

Z:\>dmd D0802.d

Z:\>D0802
0.01    48.45620     48.35853     0.48622     0.48573
0.02    48.12690     47.93224     0.96912     0.96717
0.03    47.80238     47.51136     1.44876     1.44437
0.04    47.48254     47.09579     1.92517     1.91740
0.05    47.16726     46.68541     2.39841     2.38629
0.06    46.85647     46.28011     2.86852     2.85111
0.07    46.55007     45.87980     3.33554     3.31189
0.08    46.24796     45.48436     3.79952     3.76870
0.09    45.95006     45.09369     4.26050     4.22158
省略
6.20     9.25067    -23.74819   125.75002     2.41403
6.21     9.22711    -23.78559   125.84241     2.17636
6.22     9.20359    -23.82283   125.93456     1.93832
6.23     9.18010    -23.85991   126.02648     1.69991
6.24     9.15665    -23.89683   126.11816     1.46112
6.25     9.13323    -23.93360   126.20961     1.22197
6.26     9.10985    -23.97021   126.30083     0.98245
6.27     9.08651    -24.00667   126.39181     0.74256
6.28     9.06321    -24.04297   126.48256     0.50232
6.29     9.03994    -24.07911   126.57307     0.26171
6.30     9.01671    -24.11510   126.66335     0.02073
6.31     8.99351    -24.15094   126.75341    -0.22060

Go

package main

import "fmt"
import "math"

// 重力加速度
const g float64 = -9.8
// 空気抵抗係数
const k float64 = -0.01
// 時間間隔(秒)
const h float64 = 0.01

func main() {
    // 角度
    var degree float64 = 45
    var radian float64 = degree * math.Pi / 180.0
    // 初速 250 km/h -> 秒速に変換
    var v float64 = 250 * 1000 / 3600
    // 水平方向の速度
    var vx[3] float64
    vx[0] = v * math.Cos(radian)
    // 鉛直方向の速度
    var vy[3] float64
    vy[0] = v * math.Sin(radian)
    // 経過秒数
    var t float64 = 0.0
    // 位置
    var x[3] float64
    x[0] = 0.0
    var y[3] float64
    y[0] = 0.0

    // Heun法
    for i := 1; y[0] >= 0.0; i++ {
        // 経過秒数
        t = float64(i) * h

        // 位置・速度
        x[1]  =  x[0] + h *    vx[0]
        y[1]  =  y[0] + h *    vy[0]
        vx[1] = vx[0] + h * fx(vx[0], vy[0])
        vy[1] = vy[0] + h * fy(vx[0], vy[0])

        x[2]  =  x[0] + h * (  vx[0]          +    vx[1]        ) / 2
        y[2]  =  y[0] + h * (  vy[0]          +    vy[1]        ) / 2
        vx[2] = vx[0] + h * (fx(vx[0], vy[0]) + fx(vx[1], vy[1])) / 2
        vy[2] = vy[0] + h * (fy(vx[0], vy[0]) + fy(vx[1], vy[1])) / 2

        x[0]  =  x[2]
        y[0]  =  y[2]
        vx[0] = vx[2]
        vy[0] = vy[2]

        fmt.Printf("%4.2f\t%8.5f\t%9.5f\t%9.5f\t%9.5f\n", t, vx[0], vy[0], x[0], y[0])
    }
}

// 空気抵抗による水平方向の減速分
func fx(vx float64, vy float64) float64 {
    return k * math.Sqrt(vx * vx + vy * vy) * vx
}
// 重力と空気抵抗による鉛直方向の減速分
func fy(vx float64, vy float64) float64 {
    return g + (k * math.Sqrt(vx * vx + vy * vy) * vy)
}

Z:\>8g GO0802.go

Z:\>8l -o GO0802.exe GO0802.8

Z:\>GO0802
0.01    48.45620     48.35853     0.48622     0.48573
0.02    48.12690     47.93224     0.96912     0.96717
0.03    47.80238     47.51136     1.44876     1.44437
0.04    47.48254     47.09579     1.92517     1.91740
0.05    47.16726     46.68541     2.39841     2.38629
0.06    46.85647     46.28011     2.86852     2.85111
0.07    46.55007     45.87980     3.33554     3.31189
0.08    46.24796     45.48436     3.79952     3.76870
0.09    45.95006     45.09369     4.26050     4.22158
省略
6.20     9.25067    -23.74819   125.75002     2.41403
6.21     9.22711    -23.78559   125.84241     2.17636
6.22     9.20359    -23.82283   125.93456     1.93832
6.23     9.18010    -23.85991   126.02648     1.69991
6.24     9.15665    -23.89683   126.11816     1.46112
6.25     9.13323    -23.93360   126.20961     1.22197
6.26     9.10985    -23.97021   126.30083     0.98245
6.27     9.08651    -24.00667   126.39181     0.74256
6.28     9.06321    -24.04297   126.48256     0.50232
6.29     9.03994    -24.07911   126.57307     0.26171
6.30     9.01671    -24.11510   126.66335     0.02073
6.31     8.99351    -24.15094   126.75341    -0.22060

Scala

object Scala0802 {

    // 重力加速度
    val g = -9.8
    // 空気抵抗係数
    val k = -0.01
    // 時間間隔(秒)
    val h = 0.01

    def main(args: Array[String]) {
        // 角度
        val degree = 45
        val radian = degree * Math.PI / 180.0
        // 初速 250 km/h -> 秒速に変換
        val v = 250 * 1000 / 3600
        // 水平方向の速度
        val vx = v * Math.cos(radian)
        // 鉛直方向の速度
        val vy = v * Math.sin(radian)
        // 位置
        val x = 0.0
        val y = 0.0

        // Heun法
        heun(1, vx, vy, x, y)
    }

    // Heun法
    def heun(i:Int, vx:Double, vy:Double, x:Double, y:Double):Unit = {
        // 経過秒数
        val t = i * h

        // 位置・速度
        val wx2  =  x + h *    vx
        val wy2  =  y + h *    vy
        val wvx2 = vx + h * fx(vx, vy)
        val wvy2 = vy + h * fy(vx, vy)

        val wx   =  x + h * (   vx      +    wvx2       ) / 2
        val wy   =  y + h * (   vy      +    wvy2       ) / 2
        val wvx  = vx + h * (fx(vx, vy) + fx(wvx2, wvy2)) / 2
        val wvy  = vy + h * (fy(vx, vy) + fy(wvx2, wvy2)) / 2

        println("%4.2f\t%8.5f\t%9.5f\t%9.5f\t%8.5f".format(t, wvx, wvy, wx, wy))

        if (wy >= 0.0)
            heun(i+1, wvx, wvy, wx, wy)
        else
            ()
    }

    // 空気抵抗による水平方向の減速分
    def fx(vx:Double, vy:Double) = {
        k * Math.sqrt(vx * vx + vy * vy) * vx
    }
    // 重力と空気抵抗による鉛直方向の減速分
    def fy(vx:Double, vy:Double) = {
        g + (k * Math.sqrt(vx * vx + vy * vy) * vy)
    }
}

Z:\>scala Scala0802.scala
0.01    48.45620     48.35853     0.48622    0.48573
0.02    48.12690     47.93224     0.96912    0.96717
0.03    47.80238     47.51136     1.44876    1.44437
0.04    47.48254     47.09579     1.92517    1.91740
0.05    47.16726     46.68541     2.39841    2.38629
0.06    46.85647     46.28011     2.86852    2.85111
0.07    46.55007     45.87980     3.33554    3.31189
0.08    46.24796     45.48436     3.79952    3.76870
0.09    45.95006     45.09369     4.26050    4.22158
省略
6.20     9.25067    -23.74819   125.75002    2.41403
6.21     9.22711    -23.78559   125.84241    2.17636
6.22     9.20359    -23.82283   125.93456    1.93832
6.23     9.18010    -23.85991   126.02648    1.69991
6.24     9.15665    -23.89683   126.11816    1.46112
6.25     9.13323    -23.93360   126.20961    1.22197
6.26     9.10985    -23.97021   126.30083    0.98245
6.27     9.08651    -24.00667   126.39181    0.74256
6.28     9.06321    -24.04297   126.48256    0.50232
6.29     9.03994    -24.07911   126.57307    0.26171
6.30     9.01671    -24.11510   126.66335    0.02073
6.31     8.99351    -24.15094   126.75341   -0.22060

F#

module Fs0802

open System

// 重力加速度
let g = -9.8
// 空気抵抗係数
let k = -0.01
// 時間間隔(秒)
let h = 0.01

// 角度
let degree = 45.0
let radian = degree * Math.PI / 180.0
// 初速 250 km/h -> 秒速に変換
let v = float(250 * 1000 / 3600)
// 水平方向の速度
let vx = v * Math.Cos(radian)
// 鉛直方向の速度
let vy = v * Math.Sin(radian)
// 位置
let x = 0.0
let y = 0.0

// 空気抵抗による水平方向の減速分
let fx(vx:Double) (vy:Double) =
    k * Math.Sqrt(vx * vx + vy * vy) * vx
// 重力と空気抵抗による鉛直方向の減速分
let fy(vx:Double) (vy:Double) =
    g + (k * Math.Sqrt(vx * vx + vy * vy) * vy)

// Heun法
let rec heun(i:int) (vx:double) (vy:double) (x:double) (y:double):unit =
    // 経過秒数
    let t = float(i) * h

    // 位置・速度
    let wx2  =  x + h *     vx
    let wy2  =  y + h *     vy
    let wvx2 = vx + h * (fx vx vy)
    let wvy2 = vy + h * (fy vx vy)

    let wx   =  x + h * (    vx     +     wvx2      ) / 2.0
    let wy   =  y + h * (    vy     +     wvy2      ) / 2.0
    let wvx  = vx + h * ((fx vx vy) + (fx wvx2 wvy2)) / 2.0
    let wvy  = vy + h * ((fy vx vy) + (fy wvx2 wvy2)) / 2.0

    printfn "%4.2f\t%8.5f\t%9.5f\t%9.5f\t%8.5f" t wvx wvy wx wy

    if wy >= 0.0 then
        (heun (i+1) wvx wvy wx wy)
    else
        ()

// Heun法
(heun 1 vx vy x y)

exit 0

Z:\>fsi  --nologo --quiet Fs0802.fs
0.01    48.45620     48.35853     0.48622    0.48573
0.02    48.12690     47.93224     0.96912    0.96717
0.03    47.80238     47.51136     1.44876    1.44437
0.04    47.48254     47.09579     1.92517    1.91740
0.05    47.16726     46.68541     2.39841    2.38629
0.06    46.85647     46.28011     2.86852    2.85111
0.07    46.55007     45.87980     3.33554    3.31189
0.08    46.24796     45.48436     3.79952    3.76870
0.09    45.95006     45.09369     4.26050    4.22158
省略
6.20     9.25067    -23.74819   125.75002    2.41403
6.21     9.22711    -23.78559   125.84241    2.17636
6.22     9.20359    -23.82283   125.93456    1.93832
6.23     9.18010    -23.85991   126.02648    1.69991
6.24     9.15665    -23.89683   126.11816    1.46112
6.25     9.13323    -23.93360   126.20961    1.22197
6.26     9.10985    -23.97021   126.30083    0.98245
6.27     9.08651    -24.00667   126.39181    0.74256
6.28     9.06321    -24.04297   126.48256    0.50232
6.29     9.03994    -24.07911   126.57307    0.26171
6.30     9.01671    -24.11510   126.66335    0.02073
6.31     8.99351    -24.15094   126.75341   -0.22060

Clojure

; 重力加速度
(def g -9.8)
; 空気抵抗係数
(def k -0.01)
; 時間間隔(秒)
(def h 0.01)

; 角度
(def degree 45.0)
(def radian (* degree (/ (. Math PI) 180.0)))
; 初速 250 km/h -> 秒速に変換
(def v (quot (* 250 1000) 3600))
; 水平方向の速度
(def vx (* v (. Math cos radian)))
; 鉛直方向の速度
(def vy (* v (. Math sin radian)))
; 位置
(def x 0.0)
(def y 0.0)

; 空気抵抗による水平方向の減速分
(defn fx[vx vy]
    (* k (* (. Math sqrt (+ (* vx vx) (* vy vy))) vx)))
; 重力と空気抵抗による鉛直方向の減速分
(defn fy[vx vy]
    (+ g (* k (* (. Math sqrt (+ (* vx vx) (* vy vy))) vy))))

; Heun法
(defn heun[i vx vy x y]
    ; 経過秒数
    (def t (* i h))

    ; 位置・速度
    (def wx2  (+  x (* h     vx                             )))
    (def wy2  (+  y (* h     vy                             )))
    (def wvx2 (+ vx (* h (fx vx vy)                         )))
    (def wvy2 (+ vy (* h (fy vx vy)                         )))

    (def wx   (+  x (* h (/ (+     vx         wvx2      ) 2))))
    (def wy   (+  y (* h (/ (+     vy         wvy2      ) 2))))
    (def wvx  (+ vx (* h (/ (+ (fx vx vy) (fx wvx2 wvy2)) 2))))
    (def wvy  (+ vy (* h (/ (+ (fy vx vy) (fy wvx2 wvy2)) 2))))

    (println (format "%4.2f\t%8.5f\t%9.5f\t%9.5f\t%9.5f" t wvx wvy wx wy))

    (if (>= wy 0.0)
        (heun (+ i 1) wvx wvy wx wy)
        nil))

(heun 1 vx vy x y)

Z:\>java -cp C:\ProgramFiles\clojure-1.5.1\clojure-1.5.1.jar clojure.main Clj0802.clj
0.01    48.45620     48.35853     0.48622     0.48573
0.02    48.12690     47.93224     0.96912     0.96717
0.03    47.80238     47.51136     1.44876     1.44437
0.04    47.48254     47.09579     1.92517     1.91740
0.05    47.16726     46.68541     2.39841     2.38629
0.06    46.85647     46.28011     2.86852     2.85111
0.07    46.55007     45.87980     3.33554     3.31189
0.08    46.24796     45.48436     3.79952     3.76870
0.09    45.95006     45.09369     4.26050     4.22158
省略
6.20     9.25067    -23.74819   125.75002     2.41403
6.21     9.22711    -23.78559   125.84241     2.17636
6.22     9.20359    -23.82283   125.93456     1.93832
6.23     9.18010    -23.85991   126.02648     1.69991
6.24     9.15665    -23.89683   126.11816     1.46112
6.25     9.13323    -23.93360   126.20961     1.22197
6.26     9.10985    -23.97021   126.30083     0.98245
6.27     9.08651    -24.00667   126.39181     0.74256
6.28     9.06321    -24.04297   126.48256     0.50232
6.29     9.03994    -24.07911   126.57307     0.26171
6.30     9.01671    -24.11510   126.66335     0.02073
6.31     8.99351    -24.15094   126.75341    -0.22060

Haskell

import Text.Printf

-- 重力加速度
g = -9.8 :: Double
-- 空気抵抗係数
k = -0.01 :: Double
-- 時間間隔(秒)
h = 0.01 :: Double

-- 空気抵抗による水平方向の減速分
fx::Double->Double->Double
fx vx vy =
    let
        v = sqrt(vx * vx + vy * vy)
    in
        k * v * vx

-- 空気抵抗による鉛直方向の減速分
fy::Double->Double->Double
fy vx vy =
    let
        v = sqrt(vx * vx + vy * vy)
    in
        g + (k * v * vy)

-- Heun法
heun::Integer->Double->Double->Double->Double->IO ()
heun i vx vy x y =
    let
        t = (fromIntegral i) * h

        wx2  =  x + h *     vx
        wy2  =  y + h *     vy
        wvx2 = vx + h * (fx vx vy)
        wvy2 = vy + h * (fy vx vy)

        wx   =  x + h * (    vx     +     wvx2      ) / 2.0
        wy   =  y + h * (    vy     +     wvy2      ) / 2.0
        wvx  = vx + h * ((fx vx vy) + (fx wvx2 wvy2)) / 2.0
        wvy  = vy + h * ((fy vx vy) + (fy wvx2 wvy2)) / 2.0
    in
        if y >= 0.0
            then do
                printf "%4.2f\t%8.5f\t%9.5f\t%9.5f\t%8.5f\n" t wvx wvy wx wy
                heun (i+1) wvx wvy wx wy
            else
                return ()

main = do
    -- 角度
    let degree = 45.0 :: Double
    let radian = degree * pi / 180.0
    -- 初速 250 km/h -> 秒速に変換
    let v = (fromIntegral (250 * 1000 `div` 3600))

    -- 水平方向の速度
    let vx = v * cos(radian)
    -- 鉛直方向の速度
    let vy = v * sin(radian)
    -- 位置
    let x = 0.0
    let y = 0.0

    -- Heun法
    heun 1 vx vy x y

Z:\>runghc Hs0802.hs
0.01    48.45620     48.35853     0.48622    0.48573
0.02    48.12690     47.93224     0.96912    0.96717
0.03    47.80238     47.51136     1.44876    1.44437
0.04    47.48254     47.09579     1.92517    1.91740
0.05    47.16726     46.68541     2.39841    2.38629
0.06    46.85647     46.28011     2.86852    2.85111
0.07    46.55007     45.87980     3.33554    3.31189
0.08    46.24796     45.48436     3.79952    3.76870
0.09    45.95006     45.09369     4.26050    4.22158
省略
6.20     9.25067    -23.74819   125.75002    2.41403
6.21     9.22711    -23.78559   125.84241    2.17636
6.22     9.20359    -23.82283   125.93456    1.93832
6.23     9.18010    -23.85991   126.02648    1.69991
6.24     9.15665    -23.89683   126.11816    1.46112
6.25     9.13323    -23.93360   126.20961    1.22197
6.26     9.10985    -23.97021   126.30083    0.98245
6.27     9.08651    -24.00667   126.39181    0.74256
6.28     9.06321    -24.04297   126.48256    0.50232
6.29     9.03994    -24.07911   126.57307    0.26171
6.30     9.01671    -24.11510   126.66335    0.02073
6.31     8.99351    -24.15094   126.75341   -0.22060