数学 - Python - JavaScript - 面積、体積、長さ - 積分法の応用 – 曲線の長さ - 点の運動する距離(直線軌道上、電車、速度、等加速度)

学習環境

Surface 3 (4G LTE)、Surface 3 タイプカバー、Surface ペン(端末)
Windows 10 Pro (OS)
数式入力ソフト(TeX, MathML): MathType
MathML対応ブラウザ: Firefox、Safari
MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
参考書籍

数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂和夫(著)、岩波書店)の第20章(面積、体積、長さ - 積分法の応用)、20.3(曲線の長さ)、点の運動する距離、問29.を取り組んでみる。

1. tを秒とする。
  
  v(t) = 27 - 1.8 × t = 0、t = 15となるので15秒後に停車する。
  
  走った長さについて。
  $\begin{array}{l} \int_{0}^{15} v (t) d t \\ = \int_{0}^{15} (27 - 1.8 t) d t \\ = {[27 t - \frac{1.8}{2} t^{2}]}_{0}^{15} \\ = 15 (27 - \frac{9}{10} \cdot 15) \\ = 15 (27 - \frac{9 \cdot 3}{2}) \\ = 15 \cdot \frac{54 - 27}{2} \\ = \frac{15 \cdot 27}{2} \\ = \frac{415}{2} \\ = 202.5 (m) \end{array}$
2. (1)の式を利用。
  $\begin{array}{l} 27 t - \frac{1.8}{2} t^{2} = 180 \\ 27 t - \frac{18}{20} t^{2} - 180 = 0 \\ \frac{2}{20} t^{2} - 3 t + 20 = 0 \\ t^{2} - 30 t + 200 = 0 \\ (t - 10) (t - 20) = 0 \\ t < 15 \\ t = 10 (秒) \end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, solve

print('29.')
print('(1)')
t, t0 = symbols('t t0')
v = 27 - 1.8 * t
I = Integral(v, (t, 0, 15))
for a in [I, I.doit()]:
    pprint(a)
    print()

print('(2)')
I = Integral(v, (t, 0, t0))
for t in [I, I.doit()]:
    pprint(t)
    print()

s = solve(I.doit() - 180)
pprint(s)

入出力結果(Terminal, Jupyter(IPython))

$ ./sample29.py
29.
(1)
15                 
⌠                  
⎮  (-1.8⋅t + 27) dt
⌡                  
0                  

202.500000000000

(2)
t₀                 
⌠                  
⎮  (-1.8⋅t + 27) dt
⌡                  
0                  

        2          
- 0.9⋅t₀  + 27.0⋅t₀

[10.0, 20.0]
$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.0001" value="0.01">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="0">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="250">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="250">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample29.js"></script>

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    input_r = document.querySelector('#r0'),
    input_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n',
    range = (start, end, step=1) => {
        let res = [];
        for (let i = start; i < end; i += step) {
            res.push(i);
        }
        return res;
    };

let f = (x) => 27 - 1.8 * x,
    g = (x) => 27 * x - 1.8 / 2 * x ** 2;

let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        lines = [[15, y1, 15, y2, 'red'],
                 [x1, 202.5, x2, 202.5, 'red'],
                 [10, y1, 10, y2, 'red']],
        fns = [[f, 'green'],
               [g, 'blue']],
        fns1 = [],
        fns2 = [];

    fns.forEach((o) => {
        let [fn, color] = o;
        for (let x = x1; x <= x2; x += dx) {
            let y = fn(x);

            if (Math.abs(y) < Infinity) {
                points.push([x, y, color]);
            }
        }
    });
    fns1.forEach((o) => {
        let [fn, color] = o;
        
        lines.push([x1, fn(x1), x2, fn(x2), color]);
    });
    fns2.forEach((o) => {
        let [fn, color] = o;

        for (let x = x1; x <= x2; x += dx0) {
            let g = fn(x);
            
            lines.push([x1, g(x1), x2, g(x2), color]);
        }        
    });
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);
    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', (d) => d[4] || 'black');
    
    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d) => d[2] || 'green');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);

    [fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =

Kamimura's blog

2017年10月14日土曜日

数学 - Python - JavaScript - 面積、体積、長さ - 積分法の応用 – 曲線の長さ - 点の運動する距離(直線軌道上、電車、速度、等加速度)

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