// Numbas version: finer_feedback_settings
{"name": "Dynamics - Impulse and momentum", "duration": 0, "metadata": {"description": "<p>To be used on the Mechanics wiki page under the Dynamics section, impulse and momentum page. Questions about impulse, momentum and collisions.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "allQuestions": true, "shuffleQuestions": false, "percentPass": 0, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "pickQuestions": 0, "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "showresultspage": "oncompletion", "preventleave": true, "browse": true, "showfrontpage": true}, "feedback": {"allowrevealanswer": true, "showtotalmark": true, "advicethreshold": 0, "intro": "", "feedbackmessages": [], "showanswerstate": true, "showactualmark": true, "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "type": "exam", "questions": [], "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Calculate momentum given mass and speed", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}], "functions": {}, "ungrouped_variables": ["mass1", "kg1", "velocity1", "tonnes1", "kg2", "velocity2"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>Momentum $p \\ \\mathrm{Ns} = $ mass $\\mathrm{kg} \\ \\times $ velocity $\\mathrm{ms^{-1}}$.</p>\n<p>Remember that in the above formula, the mass is measured&nbsp;in $\\mathrm{kg}$.</p>\n<h4>a)</h4>\n<p>We have</p>\n<p>\\begin{align} <br/>p &amp; = \\frac{\\var{mass1}}{1000} \\times \\var{velocity1}, \\\\<br/>&amp;= \\var{precround(kg1*velocity1,3)} \\, \\mathrm{Ns}. <br/>\\end{align}</p>\n<p>The magnitude of the momentum is $\\var{precround(kg1*velocity1,3)} \\, \\mathrm{Ns}.$</p>\n<h4>b)</h4>\n<p>We have</p>\n<p>\\begin{align} <br/>p &amp; = (\\var{tonnes1} \\times 1000) \\times \\var{velocity2}, \\\\<br/>&amp; = \\var{precround(tonnes1*1000*velocity2,3)} \\, \\mathrm{Ns}. <br/>\\end{align}</p>\n<p>The magnitude of the momentum is $\\var{precround(tonnes1*1000*velocity2,3)} \\, \\mathrm{Ns}.$</p>", "rulesets": {}, "parts": [{"precisionType": "dp", "prompt": "<p>A particle of mass $\\var{mass1} \\mathrm{g}$ moving at $\\var{velocity1} \\mathrm{ms^{-1}}$.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "kg1*velocity1", "strictPrecision": false, "minValue": "kg1*velocity1", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}, {"precisionType": "dp", "prompt": "<p>A caravan of mass $\\var{tonnes1} \\ \\mathrm{tonnes}$ moving at $\\var{velocity2} \\mathrm{ms^{-1}}$.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "kg2*velocity2", "strictPrecision": false, "minValue": "kg2*velocity2", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}], "statement": "<p>Find the magnitude of the momentum, $p&nbsp;\\ \\mathrm{Ns}$, of the following objects. Give your answers to 3 decimal places.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"kg1": {"definition": "mass1/1000", "templateType": "anything", "group": "Ungrouped variables", "name": "kg1", "description": ""}, "kg2": {"definition": "tonnes1*1000", "templateType": "anything", "group": "Ungrouped variables", "name": "kg2", "description": ""}, "mass1": {"definition": "random(200..1000#10)", "templateType": "randrange", "group": "Ungrouped variables", "name": "mass1", "description": "<p>mass in grams</p>"}, "velocity2": {"definition": "random(0.2..1.6#0.05)", "templateType": "randrange", "group": "Ungrouped variables", "name": "velocity2", "description": ""}, "velocity1": {"definition": "random(2..20#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "velocity1", "description": ""}, "tonnes1": {"definition": "random(0.5..10#0.5)", "templateType": "randrange", "group": "Ungrouped variables", "name": "tonnes1", "description": ""}}, "metadata": {"description": "<p>Find the magnitude of the momentum of some objects, given mass and speed. Apply the formula $\\rho = mv$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question"}, {"name": "Find change in speed after force applied for given time", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}], "functions": {}, "ungrouped_variables": ["mass", "force", "time", "impulse"], "tags": [], "advice": "<h4>a)</h4>\n<p>The impulse imparted is given by $\\text{impulse} = \\text{force} \\times \\text{time}$.</p>\n<p>So we have $\\var{force} \\times \\var{time} = \\var{impulse} \\mathrm{Ns}.$</p>\n<h4>b)</h4>\n<p>The Impulse-Momentum Principle states that $\\text{Impulse} = \\text{Final momentum} - \\text{Initial momentum}$.</p>\n<p>So we have</p>\n<p>\\begin{align} <br/>\\var{impulse} &amp; = \\var{mass}v - \\var{mass}u, \\\\<br/>&amp; = \\var{mass}v - 0, \\\\<br/>v &amp; = \\frac{\\var{impulse}}{\\var{mass}}, \\\\<br/>&amp; =\\var{impulse/mass} \\mathrm{ms^{-1}}. <br/>\\end{align}</p>\n<p></p>", "rulesets": {}, "parts": [{"precisionType": "sigfig", "prompt": "<p>To 3 significant figures, find the magnitude of the impulse given to the body by the force (in $\\mathrm{Ns}$).</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "force*time", "minValue": "force*time", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "sigfig", "prompt": "<p>To 3 significant figures, find the final speed of the body in $\\mathrm{ms^{-1}}$.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "impulse/mass", "minValue": "impulse/mass", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "statement": "<p>A body of mass $\\var{mass} \\mathrm{kg}$ is initially at rest on a smooth horizontal plane. Suppose that a horizontal force of magnitude $\\var{force} \\mathrm{N}$ acts on the body for $\\var{time} \\mathrm{s}$.&nbsp;</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"mass": {"definition": "random(0.5..8#0.25)", "templateType": "randrange", "group": "Ungrouped variables", "name": "mass", "description": ""}, "force": {"definition": "random(0.5..5#0.5)", "templateType": "randrange", "group": "Ungrouped variables", "name": "force", "description": ""}, "impulse": {"definition": "force*time", "templateType": "anything", "group": "Ungrouped variables", "name": "impulse", "description": ""}, "time": {"definition": "random(2..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "time", "description": ""}}, "metadata": {"description": "<p>Use the Impulse-Momentum principle to find the change in speed of an object after a constant force is applied for a given period of time.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Relate impulse to change in momentum", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}], "functions": {}, "ungrouped_variables": ["football_mass", "football_speed", "squash_mass", "squash_speed", "restitution", "squash_impulse", "squash_second_speed"], "tags": [], "advice": "<h4>a)</h4>\n<p>Impulse is change in momentum. The ball begins at rest so initially has a momentum of zero.</p>\n<p>Therefore</p>\n<p>\\begin{align} <br/>I &amp; = \\var{football_mass} \\times\\var{football_speed} - 0, \\\\<br/>&amp; = \\var{siground(football_mass*football_speed,3)} \\, \\mathrm{Ns}. <br/>\\end{align}</p>\n<p>The impulse received by the ball is $ \\var{siground(football_mass*football_speed,3)} \\, \\mathrm{Ns}$.</p>\n<h4>b)</h4>\n<p>Impulse is change in momentum. Therefore we take the rebound direction as being positive and have that&nbsp;</p>\n<p>\\begin{align} <br/>I &amp; = mv - mu, \\\\<br/>\\var{squash_impulse} &nbsp;&amp; = \\var{squash_mass} v - (\\var{squash_mass} \\times \\var{- squash_speed}), \\\\<br/>v &amp; = \\frac{\\var{squash_impulse} + (\\var{squash_mass} \\times \\var{-squash_speed})}{\\var{squash_mass}}, \\\\<br/>&amp; = \\var{siground((squash_impulse + squash_mass*-squash_speed)/squash_mass,3)} \\, \\mathrm{ms^{-1}}. <br/>\\end{align}</p>\n<p>The speed after the ball rebounds is $\\var{siground(squash_second_speed,3)} \\, \\mathrm{ms^{-1}}$.</p>", "rulesets": {}, "parts": [{"precisionType": "sigfig", "prompt": "<p>A football of mass $\\var{football_mass} \\, \\mathrm{kg}$ is at rest before&nbsp;it is kicked.</p>\n<p>What is the impulse received by the ball in $\\mathrm{Ns}$ if its speed is $\\var{football_speed}&nbsp;\\,&nbsp;\\mathrm{ms^{-1}}$ after it is struck?</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "football_mass*football_speed", "minValue": "football_mass*football_speed", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "sigfig", "prompt": "<p>A squash ball of mass $\\var{squash_mass} \\, \\mathrm{kg}$ hits a fixed vertical wall at right angles with speed $\\var{squash_speed} \\, \\mathrm{ms^{-1}}$. The ball rebounds at right angles to the wall.</p>\n<p>What is the speed of the ball in $\\mathrm{ms^{-1}}$ just after it has hit the wall, given that the magnitude of the impulse exerted by the wall on the ball is $\\var{squash_impulse} \\, \\mathrm{Ns}$?</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "(squash_impulse)/(squash_mass) - squash_speed", "minValue": "(squash_impulse)/(squash_mass) - squash_speed", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "statement": "<p>Give your&nbsp;answers to the following questions to 3 significant figures.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": "squash_second_speed>0"}, "preamble": {"css": "", "js": ""}, "variables": {"squash_impulse": {"definition": "precround(squash_mass*((1+restitution)*squash_speed),1)", "templateType": "anything", "group": "Ungrouped variables", "name": "squash_impulse", "description": ""}, "squash_mass": {"definition": "random(0.01..1#0.01)", "templateType": "randrange", "group": "Ungrouped variables", "name": "squash_mass", "description": ""}, "football_speed": {"definition": "random(10..50#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "football_speed", "description": ""}, "squash_second_speed": {"definition": "squash_impulse/squash_mass-squash_speed", "templateType": "anything", "group": "Ungrouped variables", "name": "squash_second_speed", "description": ""}, "squash_speed": {"definition": "random(0.5..5#0.5)", "templateType": "randrange", "group": "Ungrouped variables", "name": "squash_speed", "description": ""}, "football_mass": {"definition": "random(0.05..1#0.05)", "templateType": "randrange", "group": "Ungrouped variables", "name": "football_mass", "description": ""}, "restitution": {"definition": "random(0.1..0.9#0.02)", "templateType": "randrange", "group": "Ungrouped variables", "name": "restitution", "description": "<p>coefficient of restitution of the wall</p>"}}, "metadata": {"description": "<p>Finding speeds and impulses&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Impulse transmitted through a string between two objects", "extensions": [], "custom_part_types": [], "resources": [["question-resources/impulse1.png", "/srv/numbas/media/question-resources/impulse1.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}], "functions": {}, "ungrouped_variables": ["particle_speed", "A_speed", "A_mass", "B_mass", "impulse"], "tags": [], "advice": "<p>We can draw a diagram with all the speeds and impulses with arrows.</p>\n<p><img src=\"resources/question-resources/impulse1.png\" style=\"display: block; margin-left: auto; margin-right: auto;\"/></p>\n<p>Here particle $A$ has mass $m_1 = \\var{A_mass} \\, \\mathrm{kg}$ and is projected with a speed $u_1 \\, \\mathrm{ms^{-1}} = \\var{A_speed} \\, \\mathrm{ms^{-1}}$. Particle $B$ has mass $m_2 \\, \\mathrm{kg}$ to be determined and is initially at rest: $u_2 = 0 \\, \\mathrm{ms^{-1}}$. Both particles have common final velocities $v = \\var{particle_speed} \\, \\mathrm{ms^{-1}}$ and exert an impulse $I \\, \\mathrm{Ns}$ through the string as shown.</p>\n<h4>a)</h4>\n<p>To find the mass of particle $B$ we can use the principle of conservation of momentum: the total momentum in a system remains constant.</p>\n<p>\\begin{align} <br/>m_1 u_1 + m_2 u_2 &amp; = m_1 v_1 + m_2v_2, \\\\<br/>(\\var{A_mass} \\times \\var{A_speed}) + 0 &amp; = \\var{particle_speed} ( \\var{A_mass} + m_2 ), \\\\<br/>\\frac{&nbsp;\\var{A_mass} \\times \\var{A_speed}}{\\var{particle_speed}} &amp; =&nbsp;&nbsp;\\var{A_mass} + m_2, \\\\<br/>m_2 &nbsp;&amp; =&nbsp;\\frac{&nbsp;\\var{A_mass} \\times \\var{A_speed}}{\\var{particle_speed}} - \\var{A_mass}, \\\\<br/>&amp; = \\var{siground( (A_mass*A_speed)/particle_speed - A_mass,3)} \\, \\mathrm{kg}. <br/>\\end{align}</p>\n<p>So the mass of particle $B$ is $\\var{siground( (A_mass*A_speed)/particle_speed - A_mass,3)} \\, \\mathrm{kg}$.</p>\n<h4>b)</h4>\n<p>To find the impulse we consider one of the particles and apply the Impulse-Momentum Principle. Although it is easier to consider particle $B$ as it is initially at rest we will consider particle $A$ incase the mass of particle $B$ was calculated incorrectly in part <strong>a)</strong>.</p>\n<p>We will consider particle $A$ and apply the Impulse-Momentum Principle in the direction of the impulse shown in the diagram, $(\\leftarrow)$. This means our velocities will now be negative as they are acting in the opposite direction to the impulse.</p>\n<p>\\begin{align} <br/>I &amp; = mv - mu_1, \\\\<br/>&amp; = m (v - u_1), \\\\<br/>&amp; = \\var{A_mass} \\times ( - \\var{particle_speed} - ( - \\var{A_speed})), \\\\<br/>&amp; = \\var{A_mass} \\times ( \\var{ - particle_speed + A_speed} ), \\\\<br/>&amp; = \\var{siground( A_mass*(A_speed - particle_speed),3)} \\, \\mathrm{Ns}. <br/>\\end{align}</p>\n<p>The magnitude of the impulse transmitted through the string is $\\var{impulse} \\, \\mathrm{Ns}$.</p>", "rulesets": {}, "parts": [{"precisionType": "sigfig", "prompt": "<p>If the mass of particle $A$ is $\\var{A_mass} \\, \\mathrm{kg}$, what is the mass of particle $B$ in $\\mathrm{kg}$?</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "B_mass", "minValue": "B_mass", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "sigfig", "prompt": "<p>Find the magnitude in $\\mathrm{Ns}$ of the impulse transmitted through the string when it goes taut.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "impulse", "minValue": "impulse", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "statement": "<p>Two particles $A$ and $B$ are at rest and connected by a light inextensible string which is slack. Particle $A$ is projected directly away from particle $B$ with speed $\\var{A_speed} \\mathrm{ms^{-1}}$.</p>\n<p>When the string goes taut the common speed of the particles is $\\var{particle_speed} \\mathrm{ms^{-1}}$.</p>\n<p>Give your answers to the following questions to 3 significant figures.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": "impulse>0"}, "preamble": {"css": "", "js": ""}, "variables": {"particle_speed": {"definition": "random(0.5..5#0.25)", "templateType": "randrange", "group": "Ungrouped variables", "name": "particle_speed", "description": "<p>common speed of particles when the string goes taut</p>"}, "A_speed": {"definition": "random(5.1..7#0.1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "A_speed", "description": "<p>particle A is projected away from B with this speed</p>"}, "B_mass": {"definition": "(A_speed*A_mass)/particle_speed - A_mass", "templateType": "anything", "group": "Ungrouped variables", "name": "B_mass", "description": ""}, "A_mass": {"definition": "random(8..15#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "A_mass", "description": "<p>mass $A$</p>"}, "impulse": {"definition": "-A_mass*(particle_speed-A_speed)", "templateType": "anything", "group": "Ungrouped variables", "name": "impulse", "description": ""}}, "metadata": {"description": "<p>Collisions question involving principle of conservation of momentum.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Find initial speeds of particles given speeds after collision and impulse", "extensions": [], "custom_part_types": [], "resources": [["question-resources/collision.png", "/srv/numbas/media/question-resources/collision.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}], "functions": {}, "ungrouped_variables": ["A_mass", "B_mass", "A_speed", "B_speed", "impulse", "A_before_speed", "B_before_speed"], "tags": [], "advice": "<p>We can draw a diagram.</p>\n<p><img src=\"resources/question-resources/collision.png\" style=\"display: block; margin-left: auto; margin-right: auto;\"/></p>\n<p>The impulse applied to both balls is $I = \\var{impulse} \\, \\mathrm{Ns}$, and the speeds after the collision are $v_1 = \\var{A_speed} \\, \\mathrm{ms^{-1}}$ and $v_2 = \\var{B_speed} \\, \\mathrm{ms^{-1}}$ for the balls $A$ and $B$ respectively.</p>\n<h4>a)</h4>\n<p>To calculate $u_1$, the speed of ball $A$ before the collision, we use the equation $I = m_1v_1 - m_1u_1$, where $m_1$ is the mass of ball $A$, which is $\\var{A_mass} \\, \\mathrm{kg}$. We resolve in the direction of the impulse, therefore the signs of $v_1$ and $u_1$ will be reversed.</p>\n<p>\\begin{align} <br/>I &amp; = m_1v_1 - m_1u_1, \\\\<br/>\\var{impulse} &amp; = \\var{A_mass} (v_1 - u_1), \\\\<br/>\\frac{\\var{impulse}}{\\var{A_mass}} &amp; = - \\var{A_speed} - (- u_1), \\\\<br/>u_1 &amp; =&nbsp;\\frac{\\var{impulse}}{\\var{A_mass}} + \\var{A_speed}, \\\\<br/>&amp; = \\var{siground( impulse/A_mass + A_speed,3)} \\, \\mathrm{ms^{-1}}. <br/>\\end{align}</p>\n<p>The magnitude of the speed of $A$ before the collision is $\\var{siground( impulse/A_mass + A_speed,3)} \\, \\mathrm{ms^{-1}}$.</p>\n<h4>b)&nbsp;</h4>\n<p>To calculate $u_2$, the speed of ball $B$ before the collision we use the equation $I = m_2v_2 - m_2u_2$, where $m_2$ is the mass of ball $B$, which is $\\var{B_mass} \\, \\mathrm{kg}$. We resolve in the direction of the impulse shown in the diagram, which is the same as the direction of the speeds.</p>\n<p>\\begin{align} <br/>I &amp; = m_2v_2 - m_2u_2, \\\\<br/>\\var{impulse} &amp; = \\var{B_mass} ( \\var{B_speed} - u_2), \\\\<br/>\\frac{\\var{impulse} }{ \\var{B_mass}} &amp; = \\var{B_speed} - u_2, \\\\<br/>u_2 &amp; = \\var{B_speed} -&nbsp; \\frac{\\var{impulse} }{ \\var{B_mass}}, \\\\<br/>&amp; = \\var{siground( B_speed - (impulse/B_mass),3)} \\, \\mathrm{ms^{-1}}. <br/>\\end{align}</p>\n<p>If this is positive it means the direction of ball $B$ we assumed is correct; if it is negative it means the ball was originally travelling in the other direction. However we were asked to find the magnitude of the speed so we take our answer as $\\var{B_before_speed} \\, \\mathrm{ms^{-1}}$.</p>", "rulesets": {}, "parts": [{"precisionType": "sigfig", "prompt": "<p>Find the magnitude of the speed of $A$ before the collision, in $\\mathrm{ms^{-1}}$ to 3 s.f.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "A_before_speed", "minValue": "A_before_speed", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "sigfig", "prompt": "<p>Find the magnitude of the speed of $B$&nbsp;before&nbsp;the collision, in $\\mathrm{ms^{-1}}$ to 3 s.f.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "B_before_speed", "minValue": "B_before_speed", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "statement": "<p>Two balls $A$ and $B$ of mass $\\var{A_mass} \\mathrm{kg}$ and $\\var{B_mass} \\mathrm{kg}$ respectively are moving in the same straight line on a smooth horizontal surface. The balls collide. After the collision both of the balls are moving in the same direction.&nbsp;</p>\n<p>After the collision, the $A$ travels at $\\var{A_speed} \\mathrm{ms^{-1}}$ and $B$ travels at $\\var{B_speed} \\mathrm{ms^{-1}}$. The magnitude of the impulse of $A$ on $B$ is $\\var{impulse} \\mathrm{Ns}$.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"B_speed": {"definition": "random(3.1..5#0.1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "B_speed", "description": ""}, "B_mass": {"definition": "random(0.5..3#0.25)", "templateType": "randrange", "group": "Ungrouped variables", "name": "B_mass", "description": ""}, "A_mass": {"definition": "random(3.1..5#0.1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "A_mass", "description": ""}, "A_speed": {"definition": "random(0.5..3#0.1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "A_speed", "description": ""}, "impulse": {"definition": "random(5..15#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "impulse", "description": ""}, "A_before_speed": {"definition": "(impulse/A_mass)+A_speed", "templateType": "anything", "group": "Ungrouped variables", "name": "A_before_speed", "description": ""}, "B_before_speed": {"definition": "sqrt((B_speed-(impulse/B_mass))^2)", "templateType": "anything", "group": "Ungrouped variables", "name": "B_before_speed", "description": ""}}, "metadata": {"description": "<p>Two particles collide. Given the masses and final speeds of the particles, as well as the impulse imparted on each, find their initial speeds.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}], "extensions": [], "custom_part_types": [], "resources": [["question-resources/impulse1.png", "/srv/numbas/media/question-resources/impulse1.png"], ["question-resources/collision.png", "/srv/numbas/media/question-resources/collision.png"]]}