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A missile is launched straight up in the air. The height of the missile, \\(h\\) metres, above the ground \\(t\\) seconds after the launch button is pressed is given by:
\n\\(h=\\var{a}t-4.9t^2\\)
", "variablesTest": {"maxRuns": 100, "condition": ""}, "advice": "\\(h=\\var{a}t-4.9t^2\\)
\nRecall that speed is the rate of change of position with respect to time i.e. \\(v=\\frac{dh}{dt}\\)
\n\\(v=\\frac{dh}{dt}=\\var{a}-2\\times4.9t\\)
\nwhen \\(t=\\var{b}\\)
\n\\(v=\\var{a}-2\\times4.9\\times\\var{b}\\)
\n\\(v=\\simplify{{a}-9.8*{b}}m/s\\)
\n\nThe missile will reach its maximum height when its speed = 0. i.e. \\(v=\\frac{dh}{dt}=\\var{a}-2\\times4.9t=0\\)
\n\\(\\var{a}=9.8t\\)
\n\\(t=\\var{a}/9.8\\)
\nThe maximum height reached will occur when \\(t=\\simplify{{a}/9.8}\\)
\n\\(h=\\var{a}\\times\\left(\\simplify{{a}/9.8}\\right)-4.9\\times\\left(\\simplify{{a}/9.8}\\right)^2\\)
\n\\(h=\\simplify{{a}^2/19.6}\\)
\n\\(h=\\simplify{{{a}/{19.6}^0.5}^2}\\)
\n\n", "rulesets": {}, "variable_groups": [], "parts": [{"customMarkingAlgorithm": "", "marks": 0, "variableReplacements": [], "prompt": "Calculate the speed of the missile (m/s) \\(\\var{b}\\) seconds after launch. Give your answer correct to one decimal place.
\n\\(v = \\) [[0]]m/s
\nWhat is the maximum height achieved by this missile? Give your answer correct to one decimal place.
\n\\(h = \\) [[1]]m
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"}, "variables": {"b": {"name": "b", "definition": "random(3..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}, "a": {"name": "a", "definition": "random(100..300#5)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}}, "ungrouped_variables": ["a", "b"], "preamble": {"js": "", "css": ""}, "extensions": [], "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}