A cyclist rides in a straight line with constant decelaration $\\var{-{a}} \\mathrm{ms^{-2}}$ from a point $A$ to a point $B$. At point $A$ the cyclist's speed is $\\var{u}\\mathrm{ms^{-1}}$ and at point $B$ the speed is $\\var{v}\\mathrm{ms^{-1}}$.
", "advice": "a) You are given $u = \\var{u},$ $v=\\var{v}$, $a = \\var{a}$, $t = ?$, $s = ?$
\nNote that $a$ is negative because the cyclist is decelerating.
\nYou need $t$, so you use $v=u+at$, rearranged for $t$.
\n\\begin{align} t & = \\frac{v-u}{a}, \\\\
& = \\frac{\\var{v}-\\var{u}}{\\var{a}}, \\\\
& = \\var[fractionNumbers]{(v-u)/a} \\mathrm{s}. \\end{align}
The time taken to move from $A$ to $B$ is $\\var[fractionNumbers]{(v-u)/a}$ seconds.
\nb) You need the distance, $s \\mathrm{m}$. Therefore you can use $s = \\left(\\frac{u+v}{2}\\right)t$, with the $t$ value found in part a).
\n\\begin{align} s & = \\left(\\frac{u+v}{2}\\right)t, \\\\
& = \\left(\\frac{\\var{u}+\\var{v}}{2}\\right) \\times \\var[fractionNumbers]{(v-u)/a}, \\\\
& = \\var[fractionNumbers]{(u+v)*0.5*((v-u)/a)} \\mathrm{m}. \\end{align}
The distance from $A$ to $B$ is $\\var[fractionNumbers]{(u+v)*0.5*((v-u)/a)}$ metres.
c) You now have $u=\\var{u}$, $a = \\var{a}$, $t=\\var{t2}$, $s=?$ and you are trying to find the velocity at point $C$ so $v=?$
\nTherefore you can use $v = u+at$.
\n\\begin{align} v & = u + at, \\\\
& = \\var{u} + \\left(\\var{a} \\times \\var{t2}\\right), \\\\
& = \\var[fractionNumbers]{u+a*t2} \\mathrm{ms^{-1}}. \\end{align}
Note that the velocity can be negative - this means the cyclist is moving from right to left. Therefore the velocity of the cyclist at point $C$ is $\\var{-(u+a*t2)} \\mathrm{ms^{-1}}$ in the direction $\\simplify{vec:BA}.$
\nd) We now need the distance, $s \\mathrm{m}$. We have $u=\\var{u}$, $t=\\var{t2}$ and we have just evaluated the final velocity at point C, $v=\\var{u+a*t2}$.
\n\\begin{align} s & = \\left(\\frac{u + v}{2}\\right)t, \\\\
& = \\left(\\frac{\\var{u} + \\var{u+a*t2}}{2}\\right) \\times \\var{t2}, \\\\
& = \\var[fractionNumbers]{(u+u+a*t2)*0.5*t2} \\mathrm{m}. \\end{align}
The distance from A to C is $\\var[fractionNumbers]{(u+u+a*t2)*0.5*t2}$ metres.
\n", "rulesets": {}, "extensions": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-1.5 .. -0.5#0.25)", "description": "acceleration negative so that deceleration is positive
", "templateType": "randrange"}, "t2": {"name": "t2", "group": "Ungrouped variables", "definition": "random(10 .. 15#1)", "description": "", "templateType": "randrange"}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "(u + v2)*0.5*t2\n", "description": "distance A to C
\n", "templateType": "anything"}, "v2": {"name": "v2", "group": "Ungrouped variables", "definition": "u+a*t2", "description": "
velocity at point C. can be negative
", "templateType": "anything"}, "u": {"name": "u", "group": "Ungrouped variables", "definition": "random(6 .. 10#1)", "description": "initial speed
", "templateType": "randrange"}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "(v-u)/a", "description": "time from A to B
", "templateType": "anything"}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "random(1 .. 5#1)", "description": "final velocity. less than u because it is slowing down
", "templateType": "randrange"}}, "variablesTest": {"condition": "[s>0,v2<0,t2>t,abs(v2)Find the distance in $\\mathrm{m}$ from $A$ to $B$.
", "minValue": "(u+v)*0.5*t", "maxValue": "(u+v)*0.5*t", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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, "prompt": "Suppose that, after reaching $B$ the cyclist continues to move along the same straight line with constant deceleration $\\var{-a} \\mathrm{ms^{-2}}$. The cyclist is then at a point $C$ $\\var{t2}$ seconds after they leave point $A$. Find the velocity in $\\mathrm{ms^{-1}}$ of the cyclist at point $C$.
", "minValue": "u+a*t2", "maxValue": "u+a*t2", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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, "prompt": "Find the distance in $\\mathrm{m}$ from $A$ to $C$.
", "minValue": "(u+v2)*0.5*t2", "maxValue": "(u+v2)*0.5*t2", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}]}]}], "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}]}