A two part question. Students are first given the formula for the time for a ball to come to rest after being dropped on a block. Part a) asks the students to rearrange the formula to make e, the coefficient of restitution, the subject of the formula. Part b) gives students realistic values for variables in the formula and asks them to calculate the coefficient of restitution using the formula derived in part a).
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "For the first part we have \\begin{align}T&=\\frac{2v}{g}\\left(\\frac{1}{1-e}\\right)\\quad\\text{multiply both sides by }\\frac{g}{2v}\\\\\\frac{Tg}{2v}&=\\left(\\frac{1}{1-e}\\right)\\quad\\text{take reciprocals of both sides}\\\\\\frac{2v}{Tg}&=1-e\\quad\\text{add } e\\text{ and subtract }\\frac{2v}{Tg} \\text{ from both sides}\\\\e&=1-\\frac{2v}{Tg}\\end{align}
\nSubstituting the given values into this new formula gives \\begin{align}e&=1-\\frac{2\\times\\var{velocity[velocity_s]}}{\\var{time}\\times 9.8}\\\\&=\\simplify{1-(2*{velocity[velocity_s]})/({time}*9.8)}\\\\&\\approx\\simplify{{precround(1-(2*{velocity[velocity_s]})/({time}*9.8),2)}} \\end{align}
", "rulesets": {}, "extensions": [], "builtin_constants": {"e": false, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"balls": {"name": "balls", "group": "Ungrouped variables", "definition": "[\"glass\",\"chrome steel\"]", "description": "Type of ball being dropped
", "templateType": "anything", "can_override": false}, "blocks": {"name": "blocks", "group": "Ungrouped variables", "definition": "[\"brass\",\"stainless steel\",\"granite\",\"rubber\"]", "description": "Type of block ball is dropped on
", "templateType": "anything", "can_override": false}, "ball_s": {"name": "ball_s", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "used to select type of ball for the question
", "templateType": "anything", "can_override": false}, "block_s": {"name": "block_s", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "used to select type of block for the question
", "templateType": "anything", "can_override": false}, "velocity": {"name": "velocity", "group": "Ungrouped variables", "definition": "[1.74,1.93,2.17,2.43]", "description": "possible velocities of ball after first impact on block
", "templateType": "anything", "can_override": false}, "velocity_s": {"name": "velocity_s", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "selector for which value of velocity variable to use
", "templateType": "anything", "can_override": false}, "time": {"name": "time", "group": "Ungrouped variables", "definition": "switch(ball_s=0 & block_s=0,random(1.18..1.98 #0.01),\n ball_s=0 & block_s=1,random(1.87..3.54 #0.01),\n ball_s=0 & block_s=2,random(3.55..9.91 #0.01),\n ball_s=0 & block_s=3,random(0.54..0.82 #0.01),\n ball_s=1 & block_s=0,random(0.71..1.10 #0.01),\n ball_s=1 & block_s=1,random(0.96..1.55 #0.01),\n ball_s=1 & block_s=2,random(2.73..6.12 #0.01),\n random(0.61..0.93 #0.01))", "description": "time taken for ball to come to rest, depends on type of ball and block, values are constrained to give realistic value for e for the given combination.
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["balls", "blocks", "ball_s", "block_s", "velocity", "velocity_s", "time"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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": "When a ball is dropped from rest onto a horizontal surface it will bounce before eventually coming to rest after a time \\(T\\) where \\[T=\\frac{2v}{g}\\left(\\frac{1}{1-e}\\right)\\] where \\(v\\) is the speed immediately after the first impact, and \\(g\\) is the acceleration due to gravity. Transpose the formula to make \\(e\\), the coefficient of restitution, the subject. (Note: to enter a fraction such as \\(\\frac{a}{b}\\), type a/b.)
\\(e=\\)[[0]] (You must show full working on your handwritten working for this part to get full marks)
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "1-(2*v)/(T*g)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "g", "value": ""}, {"name": "t", "value": ""}, {"name": "v", "value": ""}]}], "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": "A {balls[ball_s]} ball is dropped vertically from rest onto a horizontal {blocks[block_s]} surface. The ball's speed immediately after its first impact is \\(v = \\var{velocity[velocity_s]}\\) m/s and it comes to rest after \\(T=\\var{time}\\) s. Assuming that \\(g=9.8\\) m/s\\(^2\\), estimate the coefficient of restitution for this ball/surface combination.
\n\\(e=\\)[[0]]
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1-(2*velocity[velocity_s])/(time*9.8)", "maxValue": "1-(2*velocity[velocity_s])/(time*9.8)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}]}]}], "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}, {"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}]}