// Numbas version: exam_results_page_options {"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}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["particle_speed", "A_speed", "A_mass", "B_mass", "impulse"], "name": "Impulse transmitted through a string between two objects", "tags": [], "advice": "

We can draw a diagram with all the speeds and impulses with arrows.

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.

a)

To find the mass of particle $B$ we can use the principle of conservation of momentum: the total momentum in a system remains constant.

\\begin{align}
m_1 u_1 + m_2 u_2 & = m_1 v_1 + m_2v_2, \\\\
(\\var{A_mass} \\times \\var{A_speed}) + 0 & = \\var{particle_speed} ( \\var{A_mass} + m_2 ), \\\\
\\frac{ \\var{A_mass} \\times \\var{A_speed}}{\\var{particle_speed}} & = \\var{A_mass} + m_2, \\\\
m_2 & = \\frac{ \\var{A_mass} \\times \\var{A_speed}}{\\var{particle_speed}} - \\var{A_mass}, \\\\
& = \\var{siground( (A_mass*A_speed)/particle_speed - A_mass,3)} \\, \\mathrm{kg}.
\\end{align}

So the mass of particle $B$ is $\\var{siground( (A_mass*A_speed)/particle_speed - A_mass,3)} \\, \\mathrm{kg}$.

b)

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 a).

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.

\\begin{align}
I & = mv - mu_1, \\\\
& = m (v - u_1), \\\\
& = \\var{A_mass} \\times ( - \\var{particle_speed} - ( - \\var{A_speed})), \\\\
& = \\var{A_mass} \\times ( \\var{ - particle_speed + A_speed} ), \\\\
& = \\var{siground( A_mass*(A_speed - particle_speed),3)} \\, \\mathrm{Ns}.
\\end{align}

The magnitude of the impulse transmitted through the string is $\\var{impulse} \\, \\mathrm{Ns}$.

", "rulesets": {}, "parts": [{"precisionType": "sigfig", "prompt": "

If the mass of particle $A$ is $\\var{A_mass} \\, \\mathrm{kg}$, what is the mass of particle $B$ in $\\mathrm{kg}$?

", "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": "

Find the magnitude in $\\mathrm{Ns}$ of the impulse transmitted through the string when it goes taut.

", "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}], "extensions": [], "statement": "

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 $\\var{A_speed} \\mathrm{ms^{-1}}$.

When the string goes taut the common speed of the particles is $\\var{particle_speed} \\mathrm{ms^{-1}}$.

Give your answers to the following questions to 3 significant figures.

", "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": "

common speed of particles when the string goes taut

"}, "A_speed": {"definition": "random(5.1..7#0.1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "A_speed", "description": "

particle A is projected away from B with this speed

"}, "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": "

mass $A$

"}, "impulse": {"definition": "-A_mass*(particle_speed-A_speed)", "templateType": "anything", "group": "Ungrouped variables", "name": "impulse", "description": ""}}, "metadata": {"description": "

Collisions question involving principle of conservation of momentum.

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