{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Terminal Velocity\n", "by [Paulo Marques](http://pmarques.eu), 2014/02/27\n", "\n", "---\n", "\n", "This notebook discusses the drag forces exerted on a body when traveling through air. It's also done in [Julia](julialang.org) so that I can better understand the potencial of the language.\n", "\n", "![](http://upload.wikimedia.org/wikipedia/en/8/89/Parachuters_in_hybrid_formation.jpg) \n", "\n", "---" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A falling body is subject to two forces: a downward force, $\\vec{F_g}$, due to gravity, and an upward force $\\vec{F_a}$, due to air resistance. Let's consider a body with mass $m$. Assuming an upward oriented YY axis, the following applies:\n", "\n", "$$ F_g = - g \\cdot m $$\n", "\n", "$$ F_a = - sgn(v) \\cdot {1 \\over 2} \\rho C_D A v^2 $$\n", "\n", "In these formulas, $m$ is the mass of the body, $\\rho$ is the density of air ($\\approx 1.2 Kg/m^3$), $g$ is the accelaration of gravity ($\\approx 9.8 m/s^2$), $v$ is the velocity of the body, $C_D$ is the drag coeficient of the body and $A$ is its cross-section area. $sgn(x)$ is the sign function that given a number returns +1 for positives and -1 for negatives. In the formula it makes the force always contrary to the movement of the body.\n", "\n", "Thus, the resulting force $\\vec{F}$ experienced by the body is:\n", "\n", "$$ F = - g \\cdot m - sgn(v) \\cdot {1 \\over 2} \\rho C_D A v^2 $$\n", "\n", "Given that force is directly related to accelaration, and accelaration to velocity:\n", "\n", "$$ v = \\frac{dy}{dt} $$\n", "\n", "$$ F = m \\cdot \\frac{dv}{dt} $$\n", "\n", "substituting, we get this differencial equation:\n", "\n", "$$ m \\cdot \\frac{dv}{dt} = - g \\cdot m - sgn(v) \\cdot {1 \\over 2} \\rho C_D A v^2 $$\n", "\n", "In fact, this can be expressed as a system of differential equations in the form ${\\bf f}' = {\\bf g}({\\bf f}, t)$:\n", "\n", "$$ \\begin{cases} \\frac{dy}{dt} = v \\\\\\\\ \\frac{dv}{dt} = -g - sgn(v) \\cdot {1 \\over 2} {\\rho \\over m} C_D A v^2 \\end{cases} $$\n", "\n", "---\n", "\n", "So, let's simulate this with Julia.\n", "\n", "Let's start by importing some basic libraries:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [], "source": [ "using PyPlot\n", "using ODE;" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We now define the initial conditions and constants of the problem." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# Constants \n", "g = 9.8 # Accelaration of gravity\n", "p = 1.2 # Density of air\n", "\n", "# Caracteristics of the problem\n", "m = 0.100 # A 100 g ball\n", "r = 0.10 # 10 cm radius\n", "Cd = 0.5 # Drag coeficient for a small spherical object\n", "y0 = 1000.0 # Initial height of the body (1000 m)\n", "v0 = 10.0 # Initial velocity of the body (10 m/s^2, going up)\n", "A = pi*r^2; # Cross-section area of the body;" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As said, let's define a system of ordinary differential equations in its normal form ${\\bf f}' = {\\bf g}({\\bf f}, t)$. (In the code bellow we substitute $g()$ for $gm()$ so that it doesn't clash with the acceleration of gravity constant - $g$). Note that $f$ is an $R^2$ function containing $y$ and $y'$ (and $v = y'$). " ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [], "source": [ "function gm(t, f)\n", " (y, v) = f # Extract y and v (i.e., dy/dt) from the f mapping\n", " \n", " dy_dt = v # The differential equations\n", " dv_dt = -1.0*g - sign(v)*(1./2.)*(p/m)*Cd*A*v^2.0\n", " \n", " [dy_dt; dv_dt] # Return the derivatives\n", "end;" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's define the conditions to numerically solve the problem, including a time vector:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# Initial conditions (position and velocity)\n", "start = [y0; v0]\n", "\n", "# Time span (from 0 to 5 secs)\n", "ts = [0.0; 5.0];" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now let's solve the equations numericaly and extract the corresponding $y(t)$ and $v(t)$:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [], "source": [ "t, res = ode45(gm, start, ts)\n", "y = map(x -> x[1], res)\n", "v = map(x -> x[2], res);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, we can plot the solution." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "PyPlot.Figure(PyObject )" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig, ax = subplots(1, 2, sharex=true, figsize=(10,4))\n", "\n", "ax[1][:plot](t, v)\n", "ax[1][:set_title](\"Velocity over time\");\n", "ax[1][:set_xlabel](\"Time (sec)\")\n", "ax[1][:set_ylabel](\"Velocity (m/sec)\")\n", "\n", "ax[2][:plot](t, y)\n", "ax[2][:set_title](\"Height over time\");\n", "ax[2][:set_xlabel](\"Time (sec)\")\n", "ax[2][:set_ylabel](\"Height (m)\");\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As you can see, the velocity starts at 10 $m/s^2$, with the ball going up. Its velocity starts decreasing, goes to zero at max height, and then becomes negative as the ball starts coming down. After a while it reaches its maximum speed: terminal velocity. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The theorerically [terminal velocity](http://en.wikipedia.org/wiki/Terminal_velocity), $V_t$, is:\n", "\n", "$$ V_t = \\sqrt{\\frac{2 m g}{\\rho A C_D}} $$\n", "\n", "Calculating it is easy enough:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "10.197118685526071" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vt = sqrt( (2.0*m*g) / (p*A*Cd) )" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, with our numerical simulation, the terminal velocity is:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "10.190666752444804" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# The terminal velocity\n", "vt_numeric = abs(minimum(v))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, that's what I call pretty neat!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "---\n", "\n", "# MIT LICENSE\n", "\n", "> Copyright (C) 2014 Paulo Marques (pjp.marques@gmail.com)\n", ">\n", "> Permission is hereby granted, free of charge, to any person obtaining a copy of \n", "> this software and associated documentation files (the \"Software\"), to deal in\n", "> the Software without restriction, including without limitation the rights to\n", "> use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of\n", "> the Software, and to permit persons to whom the Software is furnished to do so,\n", "> subject to the following conditions:\n", "> \n", "> The above copyright notice and this permission notice shall be included in all \n", "> copies or substantial portions of the Software.\n", "> \n", "> THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n", "> IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS\n", "> FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR\n", "> COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER\n", "> IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN \n", "> CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE." ] } ], "metadata": { "kernelspec": { "display_name": "Julia 0.4.3", "language": "julia", "name": "julia-0.4" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "0.4.3" } }, "nbformat": 4, "nbformat_minor": 0 }