A ball is thrown vertically upwards with a speed $\\var{u} \\mathrm{ms^{-1}}$ from a rooftop which is $\\var{s} \\mathrm{m}$ above ground. The acceleration due to gravity is $g = 9.8 \\mathrm{ms^{-2}}$.
", "metadata": {"description": "A ball is thrown vertically upwards from a position above ground level. Find its greatest height, and the total time its in the air.
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", "definition": "-9.8", "group": "Ungrouped variables", "templateType": "number", "name": "a"}, "s": {"description": "distance of rooftop above ground
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", "definition": "precround(t,3)", "group": "Ungrouped variables", "templateType": "anything", "name": "t2"}}, "parts": [{"marks": 1, "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "maxValue": "max_height", "minValue": "max_height", "type": "numberentry", "correctAnswerFraction": false, "customMarkingAlgorithm": "", "precision": "3", "allowFractions": false, "showFeedbackIcon": true, "strictPrecision": false, "showCorrectAnswer": true, "prompt": "Find the greatest height in $\\mathrm{m}$ (to 3d.p.) above the ground reached by the ball.
", "precisionPartialCredit": 0, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "precisionMessage": "You have not given your answer to the correct precision.
", "variableReplacements": [], "unitTests": [], "mustBeReducedPC": 0, "mustBeReduced": false, "correctAnswerStyle": "plain"}, {"marks": 1, "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "maxValue": "t2", "minValue": "t2", "type": "numberentry", "correctAnswerFraction": false, "customMarkingAlgorithm": "", "precision": "3", "allowFractions": false, "showFeedbackIcon": true, "strictPrecision": true, "showCorrectAnswer": true, "prompt": "The time in $\\mathrm{s}$ (to 3d.p.) of flight of the ball, before it lands on the ground.
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", "variableReplacements": [], "unitTests": [], "mustBeReducedPC": 0, "mustBeReduced": false, "correctAnswerStyle": "plain"}], "preamble": {"css": "", "js": ""}, "advice": "You can model the ball as a particle moving in a straight line with constant acceleration of magnitude $9.8 \\mathrm{ms}^{-2}$ due to gravity. As the ball is thrown upwards, we take the positive direction as pointing upwards.
\nWe are told $u=\\var{u}$ and $a=\\var{a}$ and we are asked to find the greatest height, $s$, reached by the ball. At its greatest height, the ball will for an instant have a velocity of $v=0$ before it starts to fall back down.
\nTherefore we can use the equation $v^2=u^2+2as$, rearranged for $s$.
\nNote that the rooftop is $\\var{s} \\mathrm{m}$ above ground level, so the distance travelled by the ball to its greatest height is $(\\simplify{s-{s}}) \\mathrm{m}$.
\n\\begin{align} v^2 & =u^2+2a(s-\\var{s}), \\\\
2a(s-\\var{s}) & = v^2-u^2, \\\\
s-\\var{s} & = \\frac{v^2-u^2}{2a}, \\\\
& = \\frac{0 - \\var{u}^2}{2 \\times \\var{a}}, \\\\
& = \\var{precround(-u^2/(2*a),3)}, \\\\
s &= \\simplify[]{{s}+{precround(-u^2/(2*a),3)}}, \\\\
&= \\var{precround(max_height,3)} \\mathrm{m}.
\\end{align}
The greatest height reached by the ball is $\\var{precround(max_height,3)} \\mathrm{m}$.
\nThe flight time of the ball is the length of time between it being thrown from the rooftop and when it hits the ground below.
\nHere, the rooftop is $\\var{s}\\mathrm{m}$ above the ground, therefore the ball will stop moving when it hits the ground at a height $\\var{-s} \\mathrm{m}$ from its original position.
\nTherefore we can use the equation $s=ut+\\frac{1}{2}at^2$ rearranged for $t$.
\n\\begin{align} s & = ut + \\frac{1}{2}at^2, \\\\
-\\var{s} & = \\var{u}t - \\frac{49}{10}t^2, \\\\
\\frac{49}{10}t^2 - \\var{u}t - \\var{s} &= 0.
\\end{align}
Using the quadratic formula, taking only the positive term as we are modelling time, this gives
\n\\begin{align} t & = \\frac{u + \\sqrt{u^2 +19.6s)}}{9.8}, \\\\
& = \\frac{\\var{u} + \\sqrt{\\var{u^2} + \\left(19.6 \\times \\var{s}\\right)}}{9.8}, \\\\
& = \\var{precround((u+ (u^2 + 19.6*s)^(1/2))/9.8,3)} \\mathrm{s}.
\\end{align}
The flight time of the ball is $\\var{precround((u+ (u^2 + 19.6*s)^(1/2))/9.8,3)}$ seconds.
", "type": "question", "contributors": [{"name": "John Bridges", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/913/"}]}]}], "contributors": [{"name": "John Bridges", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/913/"}]}