jonathan-taylor · September 18, 2019 23:33
diff --git a/Lab writeup.ipynb b/Lab writeup.ipynb
 {
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Relationship between launch angle, horizontal range (and vertical starting point)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we start at height $y_0$ e.g. $97.0 cm=0.97m$, then the time until we hit the ground is found by setting $y_t=0$ in the equation\n",
    "$$\n",
    "y_t = y_0 + v_0 \\sin \\theta \\cdot t - \\frac{g}{2}t^2\n",
    "$$\n",
    "and solving for $t$.\n",
    "\n",
    "Here, the speed (magnitude of initial velocity) is $v_0$ so its vertical component is $v_0 \\sin \\theta$ where\n",
    "$\\theta$ is the angle with the horizon at launch.\n",
    "\n",
    "So, \n",
    "$$\n",
    "\\Delta_y = v_0 \\sin \\theta \\cdot t - \\frac{g}{2} t^2\n",
    "$$\n",
    "And\n",
    "$$\n",
    "t = \\frac{v_0 \\sin \\theta \\pm \\sqrt{v_0^2 \\sin^2 \\theta - 2 g \\Delta_y}}{g}\n",
    "$$\n",
    "Here, $\\Delta_y < 0$ so the square root will always be OK. We always want the positive time, so\n",
    "$$\n",
    "t = \\frac{\\sqrt{v_0^2 \\sin^2 \\theta - 2 g \\Delta_y} + v_0 \\sin \\theta}{g}.\n",
    "$$\n",
    "\n",
    "If $\\Delta_y=0$ then this becomes\n",
    "$$\n",
    "\\frac{2 \\cdot v_0 \\sin \\theta}{g}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The horizontal distance traveled will be\n",
    "$$\n",
    "v_0 \\cos \\theta \\cdot \\frac{\\sqrt{v_0^2 \\sin^2 \\theta - 2 g \\Delta_y} + v_0 \\sin \\theta}{g} = \n",
    "v_0^2 \\cos \\theta \\cdot \\frac{\\sqrt{\\sin^2 \\theta - 2 g \\Delta_y / v_0^2} + \\sin \\theta}{g}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x11845bdd8>]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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YehAzr3o3JyfH5ebmerJtkUD03oJNXDN1Hjk92vH8pGHEROi+wUBkZnnOuZz9LdPtZiIB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\nRK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np, matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "theta = np.linspace(0, np.pi / 2, 101)\n",
    "v_0 = 2.20\n",
    "delta_y = -0.97\n",
    "g = 9.8\n",
    "plt.plot(theta * 360 / (2 * np.pi), v_0**2 * np.cos(theta) * (np.sqrt(np.sin(theta)**2 - 2 * g * delta_y / v_0**2) + np.sin(theta)) / g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "235.5589098293673"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# import numpy as np;\n",
    "204 * np.sqrt(1 + np.tan(30 * (2 * np.pi) / 360)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.7596879128588212"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.cos(15)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "219.4565068527247"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sqrt(212**2+56.72**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "jupytext": {
   "cell_metadata_filter": "all,-slideshow"
  },
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
 }
	{
	"cells": [
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"## Relationship between launch angle, horizontal range (and vertical starting point)"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"If we start at height $y_0$ e.g. $97.0 cm=0.97m$, then the time until we hit the ground is found by setting $y_t=0$ in the equation\n",
	"$$\n",
	"y_t = y_0 + v_0 \\sin \\theta \\cdot t - \\frac{g}{2}t^2\n",
	"$$\n",
	"and solving for $t$.\n",
	"\n",
	"Here, the speed (magnitude of initial velocity) is $v_0$ so its vertical component is $v_0 \\sin \\theta$ where\n",
	"$\\theta$ is the angle with the horizon at launch.\n",
	"\n",
	"So, \n",
	"$$\n",
	"\\Delta_y = v_0 \\sin \\theta \\cdot t - \\frac{g}{2} t^2\n",
	"$$\n",
	"And\n",
	"$$\n",
	"t = \\frac{v_0 \\sin \\theta \\pm \\sqrt{v_0^2 \\sin^2 \\theta - 2 g \\Delta_y}}{g}\n",
	"$$\n",
	"Here, $\\Delta_y < 0$ so the square root will always be OK. We always want the positive time, so\n",
	"$$\n",
	"t = \\frac{\\sqrt{v_0^2 \\sin^2 \\theta - 2 g \\Delta_y} + v_0 \\sin \\theta}{g}.\n",
	"$$\n",
	"\n",
	"If $\\Delta_y=0$ then this becomes\n",
	"$$\n",
	"\\frac{2 \\cdot v_0 \\sin \\theta}{g}\n",
	"$$"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"The horizontal distance traveled will be\n",
	"$$\n",
	"v_0 \\cos \\theta \\cdot \\frac{\\sqrt{v_0^2 \\sin^2 \\theta - 2 g \\Delta_y} + v_0 \\sin \\theta}{g} = \n",
	"v_0^2 \\cos \\theta \\cdot \\frac{\\sqrt{\\sin^2 \\theta - 2 g \\Delta_y / v_0^2} + \\sin \\theta}{g}\n",
	"$$"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 16,
	"metadata": {},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"[<matplotlib.lines.Line2D at 0x11845bdd8>]"
	]
	},
	"execution_count": 16,
	"metadata": {},
	"output_type": "execute_result"
	},
	{
	"data": {
	"image/png": 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tYOhFhS4icpg6J0bx+7EDyF23g2dnrvY6jgpdRORInDu4G6cP6MR9Hy1nxZZy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\nRK5CYII=\n",
	"text/plain": [
	"<Figure size 432x288 with 1 Axes>"
	]
	},
	"metadata": {},
	"output_type": "display_data"
	}
	],
	"source": [
	"import numpy as np, matplotlib.pyplot as plt\n",
	"%matplotlib inline\n",
	"\n",
	"theta = np.linspace(0, np.pi / 2, 101)\n",
	"v_0 = 2.20\n",
	"delta_y = -0.97\n",
	"g = 9.8\n",
	"plt.plot(theta * 360 / (2 * np.pi), v_0*2 np.cos(theta) * (np.sqrt(np.sin(theta)*2 - 2 g * delta_y / v_0**2) + np.sin(theta)) / g)"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 12,
	"metadata": {},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"235.5589098293673"
	]
	},
	"execution_count": 12,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"# import numpy as np;\n",
	"204 * np.sqrt(1 + np.tan(30 * (2 * np.pi) / 360)**2)"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 4,
	"metadata": {},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"-0.7596879128588212"
	]
	},
	"execution_count": 4,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"np.cos(15)"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 8,
	"metadata": {},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"219.4565068527247"
	]
	},
	"execution_count": 8,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"np.sqrt(2122+56.722)"
	]
	},
	{
	"cell_type": "code",
	"execution_count": null,
	"metadata": {
	"collapsed": true
	},
	"outputs": [],
	"source": []
	}
	],
	"metadata": {
	"jupytext": {
	"cell_metadata_filter": "all,-slideshow"
	},
	"kernelspec": {
	"display_name": "Python 3",
	"language": "python",
	"name": "python3"
	},
	"language_info": {
	"codemirror_mode": {
	"name": "ipython",
	"version": 3
	},
	"file_extension": ".py",
	"mimetype": "text/x-python",
	"name": "python",
	"nbconvert_exporter": "python",
	"pygments_lexer": "ipython3",
	"version": "3.6.2"
	}
	},
	"nbformat": 4,
	"nbformat_minor": 2
	}