// Numbas version: exam_results_page_options {"name": "David's copy of Using a speed graph to find distance", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [{"name": "speeds", "variables": ["a1", "b1", "c1", "d1", "z1", "f1"]}, {"name": "acceleration", "variables": ["mab", "mbc", "mcd", "mde", "mef"]}], "functions": {}, "statement": "
You are part of an elite team analysing a high speed car race. You are given the following graph mapping the speed of one particular car as it drives around a section of the race course. The horizontal axis plots time in seconds whilst the vertical axis maps speed in metres per second ($ms^{-1}$).
\n{geogebra_applet('cecdYjwp',[[\"a1\",a1],[\"b1\",b1],[\"c1\",c1],[\"d1\",d1],[\"z1\",z1],[\"f1\",f1]])}
", "preamble": {"js": "", "css": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Use a piecewise linear graph of speed against time to find the distance travelled by a car.
\nFinally, use the total distance travelled to find the average speed.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"mbc": {"definition": "(c1-b1)/2", "group": "acceleration", "description": "", "name": "mbc", "templateType": "anything"}, "a1": {"definition": "0\n", "group": "speeds", "description": "", "name": "a1", "templateType": "anything"}, "mde": {"definition": "(z1-d1)/2", "group": "acceleration", "description": "", "name": "mde", "templateType": "anything"}, "mcd": {"definition": "(d1-c1)/2", "group": "acceleration", "description": "", "name": "mcd", "templateType": "anything"}, "c1": {"definition": "random(8,10,12,14)+b1", "group": "speeds", "description": "", "name": "c1", "templateType": "anything"}, "d2farea": {"definition": "2*z1+d1+f1", "group": "Ungrouped variables", "description": "", "name": "d2farea", "templateType": "anything"}, "mef": {"definition": "(f1-z1)/2", "group": "acceleration", "description": "", "name": "mef", "templateType": "anything"}, "f1": {"definition": "2", "group": "speeds", "description": "", "name": "f1", "templateType": "anything"}, "mab": {"definition": "(b1-a1)/2", "group": "acceleration", "description": "", "name": "mab", "templateType": "anything"}, "z1": {"definition": "c1-random(8,10,12)", "group": "speeds", "description": "", "name": "z1", "templateType": "anything"}, "d1": {"definition": "c1", "group": "speeds", "description": "", "name": "d1", "templateType": "anything"}, "area": {"definition": "d2farea+c1*2+b1+c1+b1", "group": "Ungrouped variables", "description": "", "name": "area", "templateType": "anything"}, "b1": {"definition": "random(6,8,10)+a1\n", "group": "speeds", "description": "", "name": "b1", "templateType": "anything"}}, "parts": [{"prompt": "Use the graph to calculate the distance the car travels between $4$ and $6$ seconds.
\nDistance travelled $=$ [[0]]metres
", "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "variableReplacements": [], "marks": 0, "scripts": {}, "gaps": [{"type": "numberentry", "correctAnswerFraction": false, "mustBeReducedPC": 0, "allowFractions": false, "variableReplacements": [], "mustBeReduced": false, "scripts": {}, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "minValue": "c1*2", "maxValue": "c1*2", "notationStyles": ["plain", "en", "si-en"], "marks": 1}], "showCorrectAnswer": true}, {"prompt": "Use the graph to calculate the distance the car travels between $0$ and $2$ seconds.
\nDistance travelled $=$ [[0]]metres
", "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "variableReplacements": [], "marks": 0, "scripts": {}, "gaps": [{"type": "numberentry", "correctAnswerFraction": false, "mustBeReducedPC": 0, "allowFractions": false, "variableReplacements": [], "mustBeReduced": false, "scripts": {}, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "minValue": "b1", "maxValue": "b1", "notationStyles": ["plain", "en", "si-en"], "marks": 1}], "showCorrectAnswer": true}, {"prompt": "Use the graph to calculate the distance the car travels between $2$ and $4$ seconds.
\nDistance travelled $=$ [[0]]metres
", "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "variableReplacements": [], "marks": 0, "scripts": {}, "gaps": [{"type": "numberentry", "correctAnswerFraction": false, "mustBeReducedPC": 0, "allowFractions": false, "variableReplacements": [], "mustBeReduced": false, "scripts": {}, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "minValue": "b1+c1", "maxValue": "b1+c1", "notationStyles": ["plain", "en", "si-en"], "marks": 1}], "showCorrectAnswer": true}, {"prompt": "The car travelled $\\var{area}$ metres over the $10$ second period. Calculate the average speed of the the car over the $10$ seconds in metres per second. Give your answer as a whole number or a decimal to $1$ decimal place.
\nAverage speed $=$ [[0]] ms-1
", "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "variableReplacements": [], "marks": 0, "scripts": {}, "gaps": [{"type": "numberentry", "precisionType": "dp", "correctAnswerFraction": false, "mustBeReducedPC": 0, "allowFractions": false, "strictPrecision": true, "variableReplacements": [], "mustBeReduced": false, "scripts": {}, "showCorrectAnswer": true, "maxValue": "{area/10}", "showPrecisionHint": true, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "minValue": "{area/10}", "precisionPartialCredit": 0, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "1"}], "showCorrectAnswer": true}], "extensions": ["geogebra"], "rulesets": {}, "advice": "We can use a speed graph to calculate the distance travelled in a given time interval by finding the area under the line between the start and end times.
\nThe shape made by the speed curve, the line $x=0$, and the lines $t=4$ and $t=6$ seconds is a rectangle, so we can work out the area of this section by multiplying the width by the height.
\nThe rectangle is $2$ seconds wide, and $\\var{c1}$ ms-1 high.
\n\\begin{align}
\\text{Area} &= \\text{width} \\times \\text{height}\\\\
&= 2 \\times\\var{c1}\\\\
&=\\simplify{2{c1}}\\text{.}
\\end{align}
So the distance covered in this two second interval is $\\simplify{2{c1}}$ m.
\nThe shape made by the line and $x=0$ between $0$ and $2$ seconds forms a right-angled triangle with width $2$ and height $\\var{b1}$.
\n\\begin{align}
\\text{Area}&= \\frac{1}{2}\\times \\text{width} \\times \\text{height}\\\\
&= \\frac{1}{2} \\times 2 \\times \\var{b1}\\\\
&=\\var{b1} \\text{.}
\\end{align}
So therefore,the distance covered in this two second interval, and our answer, is $\\simplify{{b1}}$ meters.
\nThe shape made by the speed curve and $x=0$ between $2$ and $4$ seconds forms a trapezium. This can be broken down in to a right angle triangle (let's call this $A$) and a rectangle (we'll call this $B$).
\nTriangle $A$ has width $2$ m and height $\\var{c1}-\\var{b1}$ ms-1.
\n\\begin{align}
A &= \\frac{1}{2}\\times \\text{width} \\times \\text{height}\\\\
&= \\frac{1}{2}\\times2 \\times\\ (\\var{c1}-\\var{b1})\\\\
&= \\var{c1}-\\var{b1}\\\\
&=\\simplify{{c1}-{b1}}\\text{.}
\\end{align}
We can work out the area of the rectangle $B$ by multiplying its width, $2$ seconds, by its height, $\\var{b1}$ ms-1:
\n\\begin{align}
B &= \\text{width} \\times \\text{height}\\\\
&= 2 \\times(\\var{c1}-\\var{b1})\\\\
&=2 \\times \\simplify{{c1}-{b1}}\\\\
&=\\simplify{2{c1-b1}}\\text{.}
\\end{align}
We can now work out the whole area under the line by adding these two areas together:
\n\\begin{align}
\\text{Area} &= A + B \\\\
&=\\simplify{{c1}-{b1}} + \\simplify{2{c1-b1}} \\\\
&=\\simplify{2{c1-b1}+{c1}-{b1}} \\text{.}
\\end{align}
The distance covered in this interval is $\\var{2(c1-b1)+c1-b1}$ m.
\nSpeed is the distance travelled per unit of time.
\n\\begin{align}
\\text{speed} &= \\frac{\\text{distance}}{\\text{time}} \\\\[0.5em]
&= \\frac{\\var{area}}{10} \\\\[0.5em]
&=\\simplify[!fractionNumbers]{{area/10}} \\text{ ms}^{-1}\\text{.}
\\end{align}