// Numbas version: finer_feedback_settings
{"name": "SUVAT equation question 11 - involves gravity", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["u"], "name": "SUVAT equation question 11 - involves gravity", "tags": [], "advice": "<p>a) We will take the upwards direction as being the positive direction, therefore gravity will act in the negative direction&nbsp;with $a=-9.8 \\mathrm{ms^{-2}}$. <br/>We have that $u=\\var{u}$ and at the greatest height above the ground the velocity of the rocket will be $v=0$ before it changes direction and heads back towards the ground. We want $s$ therefore we will use the formula $v^2=u^2+2as$, rearranged for $s$.</p>\n<p>\\begin{align} v^2 &amp;= u^2 + 2as, \\\\<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2as&nbsp;&amp;= v^2&nbsp;- u^2, \\\\<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; s &amp; = \\frac{v^2 - u^2}{2a}, \\\\<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&amp; = \\frac{0 - \\var{u}^2}{2\\times -9.8}, \\\\<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &amp; = \\var{precround(u^2/19.6,3)} \\mathrm{m}. \\end{align}</p>\n<p>The&nbsp;greatest height reached by the rocket is $\\var{precround(u^2/19.6,3)}$ metres.</p>\n<p>b) When the rocket returns to the ground, its displacement from the ground will be zero therefore $s=0$. We have $u=\\var{u}$ and $a=-9.8$ so we will use the formula $s=ut+\\frac{1}{2}at^2$ with $s=0$ to find the time it takes for the rocket to return to the ground.</p>\n<p>Therefore \\begin{align} s &amp; = ut+\\frac{1}{2}at^2, \\\\<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &amp; = t\\left(u + \\frac{1}{2}at\\right), \\end{align}</p>\n<p>setting $s=0$ and discarding the solution $t=0$ as this represents the starting position of the rocket we obtain</p>\n<p>\\begin{align} u + \\frac{1}{2}at &amp;= 0, \\\\<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;t &amp;=-\\frac{2u}{a}, \\\\<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &amp;= \\frac{2\\times \\var{u}}{9.8}, \\\\<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &amp;= \\var{precround((10/49)*u,3)} \\mathrm{s} . \\end{align}</p>\n<p>The rocket returns to the ground after&nbsp;$\\var{precround((10/49)*u,3)}$ seconds.&nbsp;</p>\n<p></p>", "rulesets": {}, "parts": [{"precisionType": "dp", "prompt": "<p>Find the greatest heightin $\\mathrm{m}$ (to 3d.p.) above the ground that is reached by the rocket.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "u^2/19.6", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "minValue": "u^2/19.6", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "<p>The total time in $\\mathrm{s}$ (to 3d.p.) before the rocket returns to the ground.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "(10/49)*u", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "minValue": "(10/49)*u", "type": "numberentry", "showPrecisionHint": false}], "extensions": [], "statement": "<p>A rocket is projected vertically upward from the ground with a speed $\\var{u} \\mathrm{ms^{-1}}$. Suppose that the acceleration due to gravity is $g= 9.8 \\mathrm{ms^{-2}}$.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"u": {"definition": "random(15..60#0.5)", "templateType": "randrange", "group": "Ungrouped variables", "name": "u", "description": ""}}, "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}]}]}], "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}]}