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{"name": "Limiting speed given acceleration function, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "b"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..7)", "description": "", "name": "n"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "b", "n"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Limiting speed given acceleration function, ", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{a} / {(b * (n -1))}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "<p>Input as a fraction or an integer and not as a decimal.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>Maximum possible speed = [[0]]$m/s$</p>\n      <p>Input as a fraction or an integer and not as a decimal.</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>An object moves in a straight line with an acceleration given by:<br/>\\[f(t)=\\frac{\\var{a}}{(1+\\var{b}t)^{\\var{n}}}\\]</p>\n  <p>where $t$ is time.</p>\n  <p>Given that the object starts from rest, find its maximum possible speed (i.e. its limiting speed).</p>", "tags": ["1st order differential equation", "acceleration and speed", "applied mathematics", "Calculus", "checked2015", "differential equation", "differential equation ", "first order differential equation", "initial conditions", "integration", "limiting value", "MAS1603", "MAS1902", "modelling", "ode"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>29/06/2012:</strong></p>\n    <p>Added, edited tags</p>\n    <p>Slight changes in display in prompt.</p>\n    <p>Added a line to explain the limit in advice (last line).</p>\n    <p>Checked calculation. No problem with checking range as variables $\\gt 0$.</p>\n    <p><strong>10/07/2012:</strong></p>\n    <p>Added requirement in prompt that numbers entered as fractions or integers. Added decimal point as forbidden string.</p>\n    <p><strong>18/07/2012:</strong></p>\n    <p>Added description.</p>\n    <p><strong>23/07/2012:</strong></p>\n    <p>Added tags.</p>\n    <p>Question appears to be working correctly.</p>\n    <p><strong>04/11/2012:</strong></p>\n    <p>Added m/s units where needed.</p>\n    <p>&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>An object moves in a straight line, acceleration given by:</p>\n<p>$\\displaystyle f(t)=\\frac{a}{(1+bt)^n}$. The object starts from rest. Find its maximum speed.&nbsp;</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>If $v(t)$ is the velocity at time $t$ then the acceleration at time $t$ is $\\displaystyle{\\frac{dv}{dt}}$.</p>\n  <p>Hence we have the differential equation for the velocity:</p>\n  <p>\\[\\frac{dv}{dt}=\\frac{\\var{a}}{(1+\\var{b}t)^{\\var{n}}}=\\simplify[std]{{a}(1+{b}t)^{-n}}\\] where $v(0)=0$ as the object starts from rest.<br/>Integrating this gives: $\\displaystyle{v(t) = \\simplify[std]{-{a}/{b*(n-1)}*(1+{b}*t)^{-n+1}+A}}$.</p>\n  <p>Since $v(0)=0$ this gives $\\displaystyle{A=\\simplify[std]{{a}/{b*(n-1)}}}$ and so the solution is:</p>\n  <p>$\\displaystyle{v(t) = \\simplify[std]{{a}/{b*(n-1)}(1-(1+{b}*t)^{-n+1})}}$</p>\n  <p>It follows that the limiting speed is:</p>\n  <p>\\[\\lim_{t \\to \\infty} v(t) = \\simplify[std]{{a}/{b*(n-1)}}m/s\\] as</p>\n  <p>\\[(1+\\var{b}t)^{\\var{-n+1}}=\\simplify[std]{1/(1+{b}t)^{n-1}} \\rightarrow 0,\\;\\;t \\rightarrow \\infty\\]</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}