Gives a plot of a velocity-time graph using JSXgraph, and a description of the motion shown in the graph. The student is asked to calculate the acceleration at different stages of the motion, and the displacement at the end of the motion.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "{graph}
\n\nThe diagram shows the velocity-time graph of the motion of a cyclist riding along a straight road. He She They accelerates uniformly from rest to \\(\\var{vmax}\\,\\mathrm{ms}^{-1}\\) in \\(\\var{t1}\\,\\mathrm{s}\\). He She They then travels at a constant velocity of \\(\\var{vmax}\\,\\mathrm{ms}^{-1}\\) for \\(\\var{t2}\\,\\mathrm{s}\\). He She They then decelerates unformly to rest in \\(\\var{t3}\\,\\mathrm{s}\\).
", "advice": "Acceleration is given by the gradient of the velocity-time graph. It can be calculated using
\\[
\\text{acceleration} = \\frac{\\text{change in velocity}}{\\text{time taken}}.
\\]
a) In the first stage of the motion, the cyclist changes velocity from \\(0\\,\\mathrm{ms}^{-1}\\) to \\(\\var{vmax}\\,\\mathrm{ms}^{-1}\\) over a period of \\(\\var{t1}\\,\\mathrm{s}\\), so the acceleration is
\\[
\\frac{\\var{vmax}}{\\var{t1}} = \\var{a1}\\,\\mathrm{ms}^{-2}.
\\]
b) In the third stage of the motion, the cyclist changes velocity from \\(\\var{vmax}\\,\\mathrm{ms}^{-1}\\) to \\(0\\,\\mathrm{ms}^{-1}\\) over a period of \\(\\var{t3}\\,\\mathrm{s}\\), so the acceleration is
\\[
-\\frac{\\var{vmax}}{\\var{t3}} = \\var{a3}\\,\\mathrm{ms}^{-2}.
\\]
Note that this is negative since the cyclist is decelerating.
c) The displacement of the cyclist from the starting point at the end of the motion is given by the area under the graph. This can be divided into two triangles and a rectangle.
The triangle for the first stage of the journey has height \\(\\var{vmax}\\) and base length \\(\\var{t1}\\), so its area is
\\[
\\frac{1}{2}\\times\\var{vmax}\\times\\var{t1}=\\var{vmax*t1/2}.
\\]
The rectangle for the second stage of the journey has height \\(\\var{vmax}\\) and base length \\(\\var{t2}\\), so its area is
\\[
\\var{vmax}\\times\\var{t2}=\\var{vmax*t2}.
\\]
The triangle for the third stage of the journey has height \\(\\var{vmax}\\) and base length \\(\\var{t3}\\), so its area is
\\[
\\frac{1}{2}\\times\\var{vmax}\\times\\var{t3}=\\var{vmax*t3/2}.
\\]
Hence the displacement after \\(\\var{t1+t2+t3}\\,\\mathrm{s}\\) is
\\[
\\var{vmax*t1/2} + \\var{vmax*t2} + \\var{vmax*t3/2} = \\var{s123}\\,\\mathrm{m}.
\\]
d) There are \\(1000\\,\\mathrm{m}\\) in \\(1\\,\\mathrm{km}\\), and \\(3600\\) seconds in an hour, so
\\[
\\var{vmax}\\,\\mathrm{ms}^{-1} = \\var{vmax} \\times \\frac{3600}{1000}\\,\\mathrm{km}\\text{ }\\mathrm{h}^{-1} = \\var{precround(vkmh,1)}\\,\\mathrm{km}\\text{ }\\mathrm{h}^{-1}.
\\]
The time taken for the first stage of the motion, in seconds.
", "templateType": "anything", "can_override": false}, "vmax": {"name": "vmax", "group": "Parameters of the motion", "definition": "random(5..10)", "description": "The maximum velocity reached during the motion, in m/s.
", "templateType": "anything", "can_override": false}, "t2": {"name": "t2", "group": "Parameters of the motion", "definition": "random(20..40)", "description": "The time taken for the second stage of the motion, in seconds.
", "templateType": "anything", "can_override": false}, "graph": {"name": "graph", "group": "Diagrams", "definition": "jsxgraph(\n 500,500,\n [-10, vmax+1, t1+t2+t3+1, -1],\n [\n \"f\": ['functiongraph',[expression(\"vmax*x/t1\"), 0, t1]],\n \"g\": ['functiongraph',[expression(\"vmax\"), t1, t1+t2]],\n \"h\": ['functiongraph',[expression(\"-vmax*x/t3+vmax+vmax*(t1+t2)/t3\"), t1+t2, t1+t2+t3]]\n ]\n)", "description": "The graph of the motion.
", "templateType": "anything", "can_override": false}, "t3": {"name": "t3", "group": "Parameters of the motion", "definition": "random(10..20)", "description": "The time taken for the third stage of the journey, in seconds.
", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Answers", "definition": "vmax/t1", "description": "The acceleration during the first stage of the journey.
", "templateType": "anything", "can_override": false}, "a3": {"name": "a3", "group": "Answers", "definition": "-vmax/t3", "description": "The acceleration during the third stage of the journey.
", "templateType": "anything", "can_override": false}, "s123": {"name": "s123", "group": "Answers", "definition": "vmax*(t1+t3)/2 + vmax*t2", "description": "The displacement at the end of the journey.
", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "Parameters of the motion", "definition": "random(1..9)", "description": "Determines the gender for the pronouns used.
", "templateType": "anything", "can_override": false}, "vkmh": {"name": "vkmh", "group": "Parameters of the motion", "definition": "vmax*36/10", "description": "The velocity in kilometres per hour.
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Parameters of the motion", "variables": ["vmax", "t1", "t2", "t3", "g", "vkmh"]}, {"name": "Diagrams", "variables": ["graph"]}, {"name": "Answers", "variables": ["a1", "a3", "s123"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "a)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Find the acceleration of the cyclist in the first \\(\\var{t1}\\,\\mathrm{s}\\) of the motion. (Give your answer to 2 decimal places where necessary.)
\n[[0]]\\(\\,\\mathrm{ms}^{-2}\\)
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Acceleration 1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a1", "maxValue": "a1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision. Please give your answer to 2 decimal places.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Find the acceleration of the cyclist in the last \\(\\var{t3}\\,\\mathrm{s}\\) of the motion. (Give your answer to 2 decimal places where necessary.)
\n[[0]]\\(\\,\\mathrm{ms}^{-2}\\)
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Acceleration 3", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a3", "maxValue": "a3", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision. Please give your answer to 2 decimal places.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Find the displacement of the cyclist from the starting point after \\(\\var{t1+t2+t3}\\,\\mathrm{s}\\).
\n[[0]]\\(\\,\\mathrm{m}\\)
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Displacement", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "s123", "maxValue": "s123", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Convert the cyclist's maximum velocity of \\(\\var{vmax}\\,\\mathrm{ms}^{-1}\\) into \\(\\mathrm{km}\\text{ }\\mathrm{h}^{-1}\\). (Give your answer to 1 decimal place where necessary.)
\n[[0]]\\(\\mathrm{km}\\text{ }\\mathrm{h}^{-1}\\)
\n", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Velocity in kmph", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "vkmh", "maxValue": "vkmh", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision. Please give your answer to 1 decimal place.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Thomas Cottrell", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1759/"}]}]}], "contributors": [{"name": "Thomas Cottrell", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1759/"}]}