We are told that the caravan is decelerating at a rate of $a= \\var{a} \\ \\mathrm{ms^{-2}}$. We know $u=\\var{u}$ and $s=\\var{s}$ and we want $v$, therefore we can use the equation $v^2=u^2+2as$.
\n\\begin{align}
v^2 & = u^2 + 2as \\\\
& = \\var{u}^2 + \\left( 2 \\times \\var{a} \\times \\var{s} \\right) \\\\
& = \\var{u^2 + 2*a*s}
\\end{align}
Therefore $v = \\sqrt{\\var{u^2 + 2*a*s}} = \\var{v}$. So the speed is $\\var{v} \\ \\mathrm{ms^{-1}}$.
\nHaving worked out the speed $v$ of the caravan after it has travelled $\\var{s} \\, \\mathrm{m}$, we can use the formula $v = u + at$ to find the time taken.
\n\\begin{align}
v &= u + at \\\\
t &= \\frac{v-u}{a} \\\\
&= \\simplify[!basic]{({v}-{u})/{a}} \\\\
&= \\var{precround((v-u)/a,3)}
\\end{align}
The caravan takes $\\var{precround((v-u)/a,3)} \\, \\mathrm{s}$ to travel $\\var{s} \\, \\mathrm{m}$.
\nTo find the tension, $T$ in Newtons, we resolve in the direction of acceleration, where $a=\\var{a}$ and $T$ is acting in the opposite direction.
\n\\begin{align}
F & = ma \\\\
- T & = \\var{mass} \\times \\var{a} \\\\
T & = \\var{precround(mass*-a,3)}.
\\end{align}
The tension in the rope is $\\var{precround(mass*-a,3)} \\ \\mathrm{N}$.
\nIf the caravan experiences a resistance to motion of magnitude $\\var{resistance}N$ this resistance will act in the opposite direction to acceleration.
\n\\begin{align} F & = ma \\\\
- T - \\var{resistance} & = \\var{mass} \\times \\var{a}\\\\
T & = \\left(\\var{mass} \\times \\var{-a}\\right) - \\var{resistance} \\\\
& = \\var{precround(mass*-a - resistance,3)}. \\end{align}
The tension in the rope is $\\var{precround(mass*-a-resistance,3)} \\ \\mathrm{N}$.
", "rulesets": {}, "parts": [{"precisionType": "dp", "prompt": "Find the speed of the caravan in $\\mathrm{ms}^{-1}$ after it has travelled $\\var{s} \\, \\mathrm{m}$.
", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "(u^2+2*a*s)^(1/2)", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "minValue": "(u^2+2*a*s)^(1/2)", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "How many seconds does the caravan take to travel $\\var{s} \\, \\mathrm{m}$?
", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [{"variable": "v", "part": "p0", "must_go_first": false}], "precision": "3", "maxValue": "(v-u)/a", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "minValue": "(v-u)/a", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "Find the tension in Newtons in the rope which pulls the caravan.
", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "-(mass*a)", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "minValue": "-(mass*a)", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "If the caravan does experience a resistance to motion of magnitude $\\var{resistance} \\, \\mathrm{N}$, what would the tension in the rope be, in Newtons?
", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "-mass*a - resistance", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "minValue": "-mass*a - resistance", "type": "numberentry", "showPrecisionHint": false}], "extensions": [], "statement": "A caravan of mass $\\var{mass} \\ \\mathrm{kg}$ is pulled on a rope by a car along a straight horizontal road. It decelerates at a constant rate of $\\var{-a} \\ \\mathrm{ms^{-2}}$ from an initial speed of $\\var{u} \\ \\mathrm{ms^{-1}}$. There is no resistance to motion.
\nGive your answers to each of the following questions to 3 decimal places.
", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "random(-5..-0.5#0.25)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a", "description": "negative because deceleration
"}, "resistance": {"definition": "random(10..30#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "resistance", "description": ""}, "s": {"definition": "random(10..30#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "s", "description": ""}, "mass": {"definition": "random(50..400#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "mass", "description": ""}, "v": {"definition": "precround((u^2+2*a*s)^(1/2),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "v", "description": "3dp
"}, "u": {"definition": "random(25..30#0.5)", "templateType": "randrange", "group": "Ungrouped variables", "name": "u", "description": "initial speed
"}}, "metadata": {"description": "A caravan pulled along by a car. Question uses SUVAT equations and $F=ma$.
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "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/"}]}