Problem 46938. Numerical computation of the optimal shooting angle of a catapult

Created by Brandon Caasenbrood

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Consider a capapult that fires a projects into the air with an initial velocity</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>v_0 \\in \\mathbb{R}_{\\ge 0}</w:t></w:r></w:customXml><w:r><w:t>. The free-flying projectile is subjected to air friction and a gravitional force. Given a desired target </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>$z_d = [x_d, y_d] \\in \\mathbb{R}^2$</w:t></w:r></w:customXml><w:r><w:t> and an initial velocity </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>v_0 \\in \\mathbb{R}_{\\ge 0}</w:t></w:r></w:customXml><w:r><w:t>, find the optimal shooting angle </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\theta^*</w:t></w:r></w:customXml><w:r><w:t>of the catapult that minimizes the distance between the target and the trajectory of the fired projectile. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>tip 1:</w:t></w:r><w:r><w:t> Consider the states </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x_1</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x_2</w:t></w:r></w:customXml><w:r><w:t> as the x- and y-position of the projectile, and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x_3</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x_4</w:t></w:r></w:customXml><w:r><w:t> as the x- and y-velocity. Then, the trajectory of the projectile can be found by solving the following ordinary differential equation (ODE):</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>        </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\dot{x_1} = x_3</w:t></w:r></w:customXml><w:r><w:t>,     </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>        </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>\\dot{x}_2 = x_4</w:t></w:r></w:customXml><w:r><w:t>      </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>        </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\dot{x}_3 = -\\nu x_3</w:t></w:r></w:customXml><w:r><w:t>,  </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>    </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>        </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\dot{x}_4 = -g - \\nu x_4</w:t></w:r></w:customXml><w:r><w:t>.   </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>where </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>g = 9.81\\; (\\text{m/s}^2)</w:t></w:r></w:customXml><w:r><w:t>, and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\nu = 0.5</w:t></w:r></w:customXml><w:r><w:t>is the friction coefficient between the air and the projectile. Use the ode45.m function to compute the trajectory of the projectile with initial conditions </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x(t = 0) = (0,0,v_0 \\cos(\\theta_k), v_0 \\sin(\\theta_k))</w:t></w:r></w:customXml><w:r><w:t>. Plotting </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x_1</w:t></w:r></w:customXml><w:r><w:t> vs. </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x_2</w:t></w:r></w:customXml><w:r><w:t> will result in the x-y trajectory of the projectile, as shown in the figure below.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>tip 2: </w:t></w:r><w:r><w:t>Use the following update law, to incrementally update the shooting angle </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\theta_k</w:t></w:r></w:customXml><w:r><w:t>:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>        </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>\\theta_{k+1} = \\theta_k + \\lambda \\, \\text{sign}(\\theta_{e,k})\\,e_k</w:t></w:r></w:customXml><w:r><w:t> </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>where </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>e_k \\in \\mathbb{R}</w:t></w:r></w:customXml><w:r><w:t> the smallest Euclidean distance between the trajectory of the projectile and the target </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>z_d</w:t></w:r></w:customXml><w:r><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\theta_{e,k} = \\text{atan2}(d_y,d_x) - \\text{atan2}(v_0\\sin(\\theta_k),v_0\\cos(\\theta_k))</w:t></w:r></w:customXml><w:r><w:t>is a difference angle, and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>$\\lambda = 0.01$</w:t></w:r></w:customXml><w:r><w:t>an update parameter.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Example of algorithm's numerical result:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[theta = catapult(25,3,25)\ntheta = \n    0.8431\n    ]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"image\"><w:customXmlPr><w:attr w:name=\"height\" w:val=\"259\"/><w:attr 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","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; "><div style="block-size: 879.667px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 406.5px 439.833px; transform-origin: 406.5px 439.833px; vertical-align: baseline; "><div style="block-size: 64.3333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 32.1667px; text-align: left; transform-origin: 383.5px 32.1667px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Consider a capapult that fires a projects into the air with an initial velocity</span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="56.5" height="20" style="width: 56.5px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">. The free-flying projectile is subjected to air friction and a gravitional force. Given a desired target </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="109" height="21" style="width: 109px; height: 21px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> and an initial velocity </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="56.5" height="20" style="width: 56.5px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, find the optimal shooting angle </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="16" height="18.5" style="width: 16px; height: 18.5px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">of the catapult that minimizes the distance between the target and the trajectory of the fired projectile. </span></span></div><div style="block-size: 42.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 21.25px; text-align: left; transform-origin: 383.5px 21.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">tip 1:</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> Consider the states </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="14.5" height="20" style="width: 14.5px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> and </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="14.5" height="20" style="width: 14.5px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> as the x- and y-position of the projectile, and </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="14.5" height="20" style="width: 14.5px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> and </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="14.5" height="20" style="width: 14.5px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> as the x- and y-velocity. Then, the trajectory of the projectile can be found by solving the following ordinary differential equation (ODE):</span></span></div><div style="block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 11px; text-align: left; transform-origin: 383.5px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">        </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="45" height="22" style="width: 45px; height: 22px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">,     </span></span></div><div style="block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 11px; text-align: left; transform-origin: 383.5px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">        </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="45.5" height="22" style="width: 45.5px; height: 22px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">      </span></span></div><div style="block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 11px; text-align: left; transform-origin: 383.5px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">        </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="63" height="22" style="width: 63px; height: 22px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">,  </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-style: italic; ">    </span></span></div><div style="block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 11px; text-align: left; transform-origin: 383.5px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">        </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="87.5" height="22" style="width: 87.5px; height: 22px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">.   </span></span></div><div style="block-size: 63.1667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 31.5833px; text-align: left; transform-origin: 383.5px 31.5833px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">where </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="97.5" height="19.5" style="width: 97.5px; height: 19.5px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, and </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="47.5" height="17.5" style="width: 47.5px; height: 17.5px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">is the friction coefficient between the air and the projectile. Use the ode45.m function to compute the trajectory of the projectile with initial conditions </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="231.5" height="20" style="width: 231.5px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">. Plotting </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="14.5" height="20" style="width: 14.5px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> vs. </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="14.5" height="20" style="width: 14.5px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> will result in the x-y trajectory of the projectile, as shown in the figure below.</span></span></div><div style="block-size: 21.8333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 10.9167px; text-align: left; transform-origin: 383.5px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">tip 2: </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Use the following update law, to incrementally update the shooting angle </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="15" height="20" style="width: 15px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">:</span></span></div><div style="block-size: 21.8333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 10.9167px; text-align: left; transform-origin: 383.5px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">        </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="153.5" height="20" style="width: 153.5px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> </span></span></div><div style="block-size: 43.6667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 21.8333px; text-align: left; transform-origin: 383.5px 21.8333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">where </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="44" height="20" style="width: 44px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> the smallest Euclidean distance between the trajectory of the projectile and the target </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="14.5" height="20" style="width: 14.5px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="298" height="20" style="width: 298px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">is a difference angle, and </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="55.5" height="17.5" style="width: 55.5px; height: 17.5px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">an update parameter.</span></span></div><div style="block-size: 20.6667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 10.3333px; text-align: left; transform-origin: 383.5px 10.3333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""></span></span></div><div style="block-size: 20.6667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 10.3333px; text-align: left; transform-origin: 383.5px 10.3333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""></span></span></div><div style="block-size: 20.6667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 10.3333px; text-align: left; transform-origin: 383.5px 10.3333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">Example of algorithm's numerical result:</span></span></div><div style="background-color: rgb(247, 247, 247); block-size: 80px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 403.5px 40px; transform-origin: 403.5px 40px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.5px 10px; transform-origin: 403.5px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">theta = catapult(25,3,25)</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.5px 10px; transform-origin: 403.5px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">theta = </span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.5px 10px; transform-origin: 403.5px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">    0.8431</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.5px 10px; transform-origin: 403.5px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">    </span></span></div></div><div style="block-size: 264.333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 383.5px 132.167px; text-align: left; transform-origin: 383.5px 132.167px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; "><img class="imageNode" width="570" height="259" style="vertical-align: baseline;width: 570px;height: 259px" 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" data-image-state="image-loaded"></div></div></div>

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Solution Stats

11 Solutions
3 Solvers

Last Solution submitted on Feb 09, 2022

Last 200 Solutions

Problem Comments

1 Comment

1 Comment

Alex on 19 Oct 2020

Nice problem. There should be much more problems involving solution of differential equations on Cody. :-)

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