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{ "metadata": { "name": "Legge_per_parabola" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "La lungezza della curva $y(x)$ pu\u00f2 essere calcolata come\n", "\n", "$$ \\ell = \\int_{x_a}^{x_b}\\sqrt{1+\\left(\\frac{dy}{dx}\\right)^2}dx$$\n", "\n", "per la nostra parabola $y(x) = \\frac{x^2}{2b}$, $\\frac{dy}{dx} = \\frac{x}{b}$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "L'intergale indefinito \u00e8 ([Vedi qui][1]):\n", "\n", "$$ \\int\\sqrt{1+\\frac{x^2}{b^2}}dx = \\frac{x}{2}\\sqrt{1+\\frac{x^2}{b^2}} + \\frac{b}{2} arcsinh\\frac{x}{b} + C$$\n", "\n", "\n", "[1]: http://www.wolframalpha.com/input/?i=intergrate+sqrt%281%2B%28x*x%29%2F%28b*b%29%29+dx" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Modulo della velocita $v$ \u00e8 sempre costante. \n", "Quindi $\\ell(t) = v\\cdot t$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Sciegliamo condtizioni iniziali cosi: \n", "\n", "$$x(0) = y(0) = 0$$ \n", "\n", "$$\\ell(0) = 0$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\\ell(t) = \\int_{0}^{x(t)}\\sqrt{1+\\frac{x^2}{b^2}}dx = \\frac{x(t)}{2}\\sqrt{1+\\frac{x^2(t)}{b^2}} + \\frac{b}{2} arcsinh\\frac{x(t)}{b}$$\n", "\n", "$$\\Rightarrow\\qquad \\frac{x(t)}{2}\\sqrt{1+\\frac{x^2(t)}{b^2}} + \\frac{b}{2} arcsinh\\frac{x(t)}{b} = v\\cdot t$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Se fossimo in grado di risolvere questa equazione -- troveremmo la legge oraria $x(t)$. \n", "Purtroppo, questo \u00e8 impossibile in modo analitico. Ma possiamo fare un calcolo numerico: " ] }, { "cell_type": "code", "collapsed": false, "input": [ "b = 0.1\n", "v = 1\n", "\n", "def l(x):\n", " return x/2*sqrt(1+x*x/(b*b))+(b/2) * np.arcsinh(x/b)\n", "\n", "x = np.linspace(0,1,100)\n", "t = l(x) / v\n", "\n", "plot(t,x)\n", "xlabel('t')\n", "ylabel('x(t)')\n", "title(\"La legge oraria\");" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "png": 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} ], "prompt_number": 1 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Ma vogliamo usare qualche funzione $x(t)$ nella [animazione][1]? \n", "\n", "Per questo possimo fare un fit con funzione $\\alpha\\sqrt{\\beta\\cdot t} + \\gamma\\sqrt[4]{\\delta\\cdot t}$:\n", "\n", "[1]: http://kkostya.github.io/esercizi/kinematics.html" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from scipy.optimize import curve_fit\n", "\n", "def xfit(t, alpha, beta, gamma, delta):\n", " return alpha*sqrt(abs(t*beta))+gamma*abs(t*delta)**(0.25)\n", "\n", "popt, __ = curve_fit(xfit, t, x)\n", "popt" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "pyout", "prompt_number": 2, "text": [ "array([ 1.31759847, -0.15162419, -0.08589974, -2.1111449 ])" ] } ], "prompt_number": 2 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Quindi la funzione $fit(t) = 1.317\\cdot\\sqrt{0.1516\\cdot t} -0.085 \\sqrt[4]{2.111\\cdot t} \\simeq x(t)$ \n", "approxima bene la legge oraria:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "plot(t,x, label='x(t)')\n", "plot(t,xfit(t,*popt), 'r', label='fit(t)')\n", "xlabel('t')\n", "ylabel('x(t)')\n", "title(\"La legge oraria e fit\");\n", "legend(loc=0);" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "png": 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CgoJwc3PTtVm/fj1dunTBwcEBgIoVK754NRNFhqLAjh0w9kuFd9nO3+ZjKV2+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\n" } ], "prompt_number": 3 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Per java script si pu\u00f2 usare \n", "\n", " 1.371*Math.sqrt(0.151*t) - 0.085*Math.pow(2.111*t,0.25)" ] } ], "metadata": {} } ] } |
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