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<h2 class="hd hd-2 unit-title">Topics and Reading List</h2>
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<p> As you go through the lesson, you will find this list of reading from the textbook complementary to the videos and exercises, and on many occasions will contain relevant material to the presentation:</p><ul><li> Acceleration: <a href="/courses/course-v1:MITx+8.01.1x+3T2018/pdfbook/0/chapter/4/9"><i class="it">chapter 4, section 4</i></a></li><li> Constant Acceleration: <a href="/courses/course-v1:MITx+8.01.1x+3T2018/pdfbook/0/chapter/4/11"><i class="it">chapter 4, section 5</i></a></li><li> 1D Kinematics and Integration: <a href="/courses/course-v1:MITx+8.01.1x+3T2018/pdfbook/0/chapter/4/16"><i class="it">chapter 4, section 6</i></a></li></ul>
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<h2 class="hd hd-2 unit-title">L2v1: Introduction to Acceleration</h2>
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<h3 class="hd hd-2">L2v1: Introduction to Acceleration</h3>
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<h2 class="hd hd-2 unit-title">L2v2: Acceleration in 1D</h2>
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<h3 class="hd hd-2">L2v2: Acceleration in 1D</h3>
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<h2 class="hd hd-2 unit-title">L2WE1/L2v3: Worked Example - Acceleration from Position</h2>
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L2WE1: Acceleration from Position (Worked Example)
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<p>
A runner travels along the x-axis, and at time [mathjaxinline]t=0[/mathjaxinline] is at the origin. The [mathjaxinline]x[/mathjaxinline]-component of the runner's position with respect to the origin is given by: </p>
<p>
[mathjaxinline]x(t)=bt^2[/mathjaxinline] </p>
<p>
where [mathjaxinline]b[/mathjaxinline] is a positive constant. </p>
<p><b class="bfseries">(Part a)</b> What are the units of the constant [mathjaxinline]b[/mathjaxinline]. Express your answer in terms of m for meter and s for seconds. </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ls_ls01_ls01_04_2_1" class="text-input-dynamath capa_inputtype inline textline">
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<div id="display_ls_ls01_ls01_04_2_1" class="equation">`{::}`</div>
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<span id="solution_ls_ls01_ls01_04_solution_1"/>
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<p><b class="bfseries">(Part b)</b> Find [mathjaxinline]v(t)[/mathjaxinline], the [mathjaxinline]x[/mathjaxinline]-component of the runner's velocity as a function of time. <p style="display:inline">[mathjaxinline]v(t) =[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ls_ls01_ls01_04_3_1" class="text-input-dynamath capa_inputtype inline textline">
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<div id="display_ls_ls01_ls01_04_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls01_ls01_04_3_1_dynamath" name="input_ls_ls01_ls01_04_3_1_dynamath"/>
</div>
</div></div> <div class="solution-span">
<span id="solution_ls_ls01_ls01_04_solution_2"/>
</div></p>
<p><b class="bfseries">(Part c)</b> Find [mathjaxinline]a(t)[/mathjaxinline], the [mathjaxinline]x[/mathjaxinline]-component of the runner's acceleration as a function of time. <p style="display:inline">[mathjaxinline]a(t) =[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ls_ls01_ls01_04_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
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<div id="display_ls_ls01_ls01_04_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls01_ls01_04_4_1_dynamath" name="input_ls_ls01_ls01_04_4_1_dynamath"/>
</div>
</div></div> <div class="solution-span">
<span id="solution_ls_ls01_ls01_04_solution_3"/>
</div></p>
<p><b class="bfseries">(Part d)</b> Do a plot of [mathjaxinline]x(t)[/mathjaxinline], [mathjaxinline]v(t)[/mathjaxinline] and [mathjaxinline]a(t)[/mathjaxinline] vs. time. Watch the video to check your answer. </p>
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<h3 class="hd hd-2">L2v3: Worked Example: Acceleration from Position</h3>
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<h2 class="hd hd-2 unit-title">L2Q1: Acceleration from Position</h2>
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L2Q1: Acceleration from Position
</h3>
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<div class="problem">
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<p>
A robot is moving along the [mathjaxinline]x[/mathjaxinline]-axis. For the time interval [mathjaxinline]t=[0\; \mathrm{s}, 3 \; \mathrm{s}][/mathjaxinline], the [mathjaxinline]x[/mathjaxinline]-component of its position as a function of time measured with respect to the origin is given by the following polynomial function: </p>
<p>
[mathjaxinline]x(t)=0.2\; t^{4}-3\; t- 5[/mathjaxinline] </p>
<p><b class="bfseries">(Part a)</b> If the [mathjaxinline]x[/mathjaxinline]-component of position, [mathjaxinline]x[/mathjaxinline], is measured in meters, what units should the three numerical constants in the expression above have? Express your answers in m for meters, s for seconds, or kg for kilograms as needed. </p>
<p>
<p style="display:inline">Units of the constant [mathjaxinline]0.2[/mathjaxinline]: </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ls_ls01_ls01_05_b_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls01_ls01_05_b_2_1" id="input_ls_ls01_ls01_05_b_2_1" aria-describedby="status_ls_ls01_ls01_05_b_2_1" value="" class="math" size="15"/>
<span class="trailing_text" id="trailing_text_ls_ls01_ls01_05_b_2_1"/>
<span class="status unanswered" id="status_ls_ls01_ls01_05_b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p id="answer_ls_ls01_ls01_05_b_2_1" class="answer"/>
<div id="display_ls_ls01_ls01_05_b_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls01_ls01_05_b_2_1_dynamath" name="input_ls_ls01_ls01_05_b_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline">Units of the constant [mathjaxinline]-3[/mathjaxinline]: </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ls_ls01_ls01_05_b_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls01_ls01_05_b_3_1" id="input_ls_ls01_ls01_05_b_3_1" aria-describedby="status_ls_ls01_ls01_05_b_3_1" value="" class="math" size="15"/>
<span class="trailing_text" id="trailing_text_ls_ls01_ls01_05_b_3_1"/>
<span class="status unanswered" id="status_ls_ls01_ls01_05_b_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls01_ls01_05_b_3_1" class="answer"/>
<div id="display_ls_ls01_ls01_05_b_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls01_ls01_05_b_3_1_dynamath" name="input_ls_ls01_ls01_05_b_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline">Units of the constant [mathjaxinline]-5[/mathjaxinline]: </p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ls_ls01_ls01_05_b_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls01_ls01_05_b_4_1" id="input_ls_ls01_ls01_05_b_4_1" aria-describedby="status_ls_ls01_ls01_05_b_4_1" value="" class="math" size="15"/>
<span class="trailing_text" id="trailing_text_ls_ls01_ls01_05_b_4_1"/>
<span class="status unanswered" id="status_ls_ls01_ls01_05_b_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls01_ls01_05_b_4_1" class="answer"/>
<div id="display_ls_ls01_ls01_05_b_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls01_ls01_05_b_4_1_dynamath" name="input_ls_ls01_ls01_05_b_4_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_ls_ls01_ls01_05_b_solution_1"/>
</div></p>
<p><b class="bfseries">(Part b)</b> What is the [mathjaxinline]x[/mathjaxinline]-component of the robot's velocity at time [mathjaxinline]t=0[/mathjaxinline]? </p>
<p>
<p style="display:inline">[mathjaxinline]v_{x} =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_ls_ls01_ls01_05_b_5_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls01_ls01_05_b_5_1" id="input_ls_ls01_ls01_05_b_5_1" aria-describedby="trailing_text_ls_ls01_ls01_05_b_5_1 status_ls_ls01_ls01_05_b_5_1" value="" size="8"/>
<span class="trailing_text" id="trailing_text_ls_ls01_ls01_05_b_5_1">[mathjaxinline]\mathrm{m\cdot s^{-1}}[/mathjaxinline]</span>
<span class="status unanswered" id="status_ls_ls01_ls01_05_b_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls01_ls01_05_b_5_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_ls_ls01_ls01_05_b_solution_2"/>
</div></p>
<p><b class="bfseries">(Part c)</b> What is [mathjaxinline]a_ x[/mathjaxinline], the [mathjaxinline]x[/mathjaxinline]-component of the robot's acceleration at time [mathjaxinline]t=3[/mathjaxinline] s? <p style="display:inline">[mathjaxinline]a_{x} =[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_ls_ls01_ls01_05_b_6_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls01_ls01_05_b_6_1" id="input_ls_ls01_ls01_05_b_6_1" aria-describedby="trailing_text_ls_ls01_ls01_05_b_6_1 status_ls_ls01_ls01_05_b_6_1" value="" size="8"/>
<span class="trailing_text" id="trailing_text_ls_ls01_ls01_05_b_6_1">[mathjaxinline]\mathrm{m\cdot s^{-2}}[/mathjaxinline]</span>
<span class="status unanswered" id="status_ls_ls01_ls01_05_b_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls01_ls01_05_b_6_1" class="answer"/>
</div>
</div></div> </p>
<p>
<div class="solution-span">
<span id="solution_ls_ls01_ls01_05_b_solution_3"/>
</div></p>
<p><b class="bfseries">(Part d)</b> What is [mathjaxinline]x_ s[/mathjaxinline], the robot's position at the instant when it is momentarily at rest? <p style="display:inline">[mathjaxinline]x_ s =[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 6" role="group"><div id="inputtype_ls_ls01_ls01_05_b_7_1" class=" capa_inputtype inline textline">
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<h2 class="hd hd-2 unit-title">L2v4: Integration</h2>
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<h2 class="hd hd-2 unit-title">List of Useful Integrals</h2>
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<p><b> List of useful integrals</b></p><ul><li><p>Integral of a polynomial function:</p><p>If \(x(t)=At^n \Longrightarrow \int_{t_{i}}^{t_{f}} x(t)dt=\frac{A}{n+1}(t_{f}^{n+1}-t_{i}^{n+1}) \)</p><p>where \(A\) and \(n\) are constants.</p></li><li><p>Integral of an exponential function:</p><p>If \(x(t)=A e^{bt}\Longrightarrow \int_{t_{i}}^{t_{f}} x(t)dt=\frac{A}{b}(e^{bt_{f}}-e^{bt_{i}}) \)</p><p>where \(A\) and \(b\) are constants.</p></li><li><p>Integral of \(1/t\):</p><p>If \(x(t)=\frac{1}{t} \Longrightarrow \int_{t_{i}}^{t_{f}} x(t)dt=\ln(t_{f})-\ln(t_{i})=\ln(\frac{t_{f}}{t_{i}}) \)</p></li><li><p>Integral of sine:</p><p>If \(x(t)=A\sin(b+ct) \Longrightarrow \int x(t)dt=-\frac{A}{c}\cos(b+ct) + D \)</p><p>where \(A\), \(b\) and \(c\) are constants, and \(D\) is an integration constant.</p></li><li><p>Integral of cosine:</p><p>If \(x(t)=A\cos(b+ct) \Longrightarrow \int x(t)dt=\frac{A}{c} \sin(b+ct) + D \)</p><p>where \(A\), \(b\) and \(c\) are constants, and \(D\) is an integration constant.</p></li></ul><p><b>External References </b></p><p><a href="http://www.wolframalpha.com/" target="blank">1. Wolfram Alpha </a></p>
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<h2 class="hd hd-2 unit-title">L2Q2: Integrating Acceleration</h2>
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L2Q2-1: Integrating Acceleration - Part I
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<p>Consider the graph below representing the \(x\)-component of acceleration as function of time for a particle moving along the \(x\)-axis. </p>
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<p> What is the meaning of the shaded area? </p>
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L2Q2-2: Integrating Acceleration - Part II
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<p>A spaceship is travelling in a straight line. The \(x\)-component of the acceleration of the spaceship at time \(t\) is known to be \(a(t)=8 t^3-2\) with the acceleration \(a\) measured in meters per second squared and \(t\) measured in seconds. </p>
<p><b class="bfseries">(Part a)</b> What is the correct expression for the \(x\)-component of the velocity \(v_{x}(t)\) of this spaceship? (Wherever \(C\) appears, it denotes a constant.)</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1">
<fieldset aria-describedby="status_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1">
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<input type="radio" name="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1" id="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1-choice_0-label" for="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1"> \(v(t)=24 \cdot t^2\)
</label>
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<div class="field">
<input type="radio" name="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1" id="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1-choice_1-label" for="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1"> \(v(t)=24 \cdot t^2-2\)
</label>
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<input type="radio" name="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1" id="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1-choice_2-label" for="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1"> \(v(t)=24 \cdot t^2-2\cdot t\)
</label>
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<input type="radio" name="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1" id="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1-choice_3-label" for="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1"> \(v(t)=24 \cdot t^2 + C\)
</label>
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<input type="radio" name="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1" id="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1-choice_4-label" for="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1"> \(v(t)=2 \cdot t^4\)
</label>
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<input type="radio" name="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1" id="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1-choice_5-label" for="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1"> \(v(t)=2 \cdot t^4-2\)
</label>
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<div class="field">
<input type="radio" name="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1" id="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1-choice_6-label" for="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1"> \(v(t)=2 \cdot t^4-2\cdot t\)
</label>
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<div class="field">
<input type="radio" name="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1" id="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1_choice_7" class="field-input input-radio" value="choice_7"/><label id="lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1-choice_7-label" for="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1"> \(v(t)=2 \cdot t^4 + C\)
</label>
</div>
<div class="field">
<input type="radio" name="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1" id="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1_choice_8" class="field-input input-radio" value="choice_8"/><label id="lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1-choice_8-label" for="input_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1_choice_8" class="response-label field-label label-inline" aria-describedby="status_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_2_1"> \(v(t)=2 \cdot t^4-2\cdot t+C\)
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</div><p><b class="bfseries">(Part b)</b> What is the change in \(x\)-component of the velocity of the spaceship between \(t_{i}=3.0\) and \(t_{f}=6.0\)?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_3_1" class=" capa_inputtype textline">
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<span class="trailing_text" id="trailing_text_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_3_1">m/s</span>
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<span id="solution_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_solution_2"/>
</div><p><b class="bfseries">(Part c)</b> It is also known that the \(x\)-component of the velocity of the spaceship at \(t=6.0\) s is \(-6.0\) m/s. Find the velocity at \(t=0\).</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_4_1" class=" capa_inputtype textline">
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<span class="trailing_text" id="trailing_text_lore_ls01_19848_bf70c1eb4ba54b4e8fca289fc7c18154_4_1">m/s</span>
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<h2 class="hd hd-2 unit-title">L2Q3: Integrating Acceleration Graph</h2>
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L2Q3-1: Integrating Acceleration Graphs, Part 1
</h3>
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<img src="/assets/courseware/v1/d9a463336addc0f55e38662580437891/asset-v1:MITx+8.01.1x+3T2018+type@asset+block/html_piecewise_acceleration_graph_checkpoint_cacbk.svg" style="float:right;width:50%"/>
<p>For a particle moving along the \(x\)-axis, the \(x\)-component of the acceleration as a function of time is given by the graph below:</p>
<p>Find the change in the \(x\)component of the velocity, \(\Delta v_x\), for the following time intervals: </p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><span> \(\Delta v_x\) from zero to 2.0 s = </span>
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<span class="trailing_text" id="trailing_text_lore_ls01_19854_1ee2d1c007714feaa9e9bd6102671327_2_1">m/s</span>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><span> \(\Delta v_x\) from 2.0 s to 4.0 s = </span>
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<span class="trailing_text" id="trailing_text_lore_ls01_19854_1ee2d1c007714feaa9e9bd6102671327_3_1">m/s</span>
<span class="status unanswered" id="status_lore_ls01_19854_1ee2d1c007714feaa9e9bd6102671327_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><span> \(\Delta v_x\) from 4.0 s to 5.0 s = </span>
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<div class="unanswered inline">
<input type="text" name="input_lore_ls01_19854_1ee2d1c007714feaa9e9bd6102671327_4_1" id="input_lore_ls01_19854_1ee2d1c007714feaa9e9bd6102671327_4_1" aria-describedby="trailing_text_lore_ls01_19854_1ee2d1c007714feaa9e9bd6102671327_4_1 status_lore_ls01_19854_1ee2d1c007714feaa9e9bd6102671327_4_1" value=""/>
<span class="trailing_text" id="trailing_text_lore_ls01_19854_1ee2d1c007714feaa9e9bd6102671327_4_1">m/s</span>
<span class="status unanswered" id="status_lore_ls01_19854_1ee2d1c007714feaa9e9bd6102671327_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><span> \(\Delta v_x\) from 5.0 s to 8.0 s = </span>
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<div class="unanswered inline">
<input type="text" name="input_lore_ls01_19854_1ee2d1c007714feaa9e9bd6102671327_5_1" id="input_lore_ls01_19854_1ee2d1c007714feaa9e9bd6102671327_5_1" aria-describedby="trailing_text_lore_ls01_19854_1ee2d1c007714feaa9e9bd6102671327_5_1 status_lore_ls01_19854_1ee2d1c007714feaa9e9bd6102671327_5_1" value=""/>
<span class="trailing_text" id="trailing_text_lore_ls01_19854_1ee2d1c007714feaa9e9bd6102671327_5_1">m/s</span>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><span> \(\Delta v_x\) from zero to 8.0 s = </span>
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<span class="trailing_text" id="trailing_text_lore_ls01_19854_1ee2d1c007714feaa9e9bd6102671327_6_1">m/s</span>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<h3 class="hd hd-3 problem-header" id="lore_ls01_19855_c20eed62fa6a4b0588847122c8abea25-problem-title" aria-describedby="block-v1:MITx+8.01.1x+3T2018+type@problem+block@lore_ls01_19855_c20eed62fa6a4b0588847122c8abea25-problem-progress" tabindex="-1">
L2Q3-2: Integrating Acceleration Graphs, Part 2
</h3>
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<p>Find \(\Delta v_x\) for the following time intervals: </p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><span> \(\Delta v_x\) from 1.0 s to 4.5 s = </span>
<div id="inputtype_lore_ls01_19855_c20eed62fa6a4b0588847122c8abea25_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
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<span class="trailing_text" id="trailing_text_lore_ls01_19855_c20eed62fa6a4b0588847122c8abea25_2_1">m/s</span>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<span id="solution_lore_ls01_19855_c20eed62fa6a4b0588847122c8abea25_solution_1"/>
</div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><span>\(\Delta v_x\) from 4.5 s to 6 s = </span>
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<span class="trailing_text" id="trailing_text_lore_ls01_19855_c20eed62fa6a4b0588847122c8abea25_3_1">m/s</span>
<span class="status unanswered" id="status_lore_ls01_19855_c20eed62fa6a4b0588847122c8abea25_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<div class="solution-span">
<span id="solution_lore_ls01_19855_c20eed62fa6a4b0588847122c8abea25_solution_2"/>
</div><p>If \(v(4) = -12\) m/s, what is \(v_x(0)\)?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><span>\(v_x(0) = \)</span>
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<span class="trailing_text" id="trailing_text_lore_ls01_19855_c20eed62fa6a4b0588847122c8abea25_4_1">m/s</span>
<span class="status unanswered" id="status_lore_ls01_19855_c20eed62fa6a4b0588847122c8abea25_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<h2 class="hd hd-2 unit-title">L2Q4: Matching Objects to Graphs</h2>
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<h3 class="hd hd-3 problem-header" id="k_w1_motion_1d_a-problem-title" aria-describedby="block-v1:MITx+8.01.1x+3T2018+type@problem+block@k_w1_motion_1d_a-problem-progress" tabindex="-1">
L2Q4-1: Graphical Representation of the Motion of Three Objects Part 1
</h3>
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<span>
<p>
Three objects, a square, a circle, and a diamond, are moving along the x-axis. The figure below shows three graphs, labeled A, B, and C, representing the position of each object as a function of time. </p>
<center>
<img src="/assets/courseware/v1/45d702fa96cfed8430ab8835559c155d/asset-v1:MITx+8.01.1x+3T2018+type@asset+block/images_K_Motion_1D_xtplot.jpg" width="550"/>
</center>
<p>
Below is an animation of the motion of the 3 objects </p>
<center>
<img src="/assets/courseware/v1/2421a5107767205b29b0fd04c9ab9e68/asset-v1:MITx+8.01.1x+3T2018+type@asset+block/images_K_Motion_1D.gif" width="550"/>
</center>
<p>
Drag and drop to match the object to the corresponding graph: </p>
<span>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_k_w1_motion_1d_a_2_1" class="capa_inputtype">
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<div class="drag_and_drop_problem_json" id="drag_and_drop_json_k_w1_motion_1d_a_2_1" style="display:none;">{"target_outline": "false", "draggables": [{"target_fields": [], "icon": "/assets/courseware/v1/31c4d71a992dc23d92d528e834832c62/asset-v1:MITx+8.01.1x+3T2018+type@asset+block/images_k_w1_motion_1d_dnd_k_w1_motion_1d_dnd_dnd_label1.png", "label": "", "can_reuse": "", "id": "A"}, {"target_fields": [], "icon": "/assets/courseware/v1/ebf543814ffab2c4fe831a88c5bc0db0/asset-v1:MITx+8.01.1x+3T2018+type@asset+block/images_k_w1_motion_1d_dnd_k_w1_motion_1d_dnd_dnd_label2.png", "label": "", "can_reuse": "", "id": "B"}, {"target_fields": [], "icon": "/assets/courseware/v1/0892656d735bb80b15acb337aa45df00/asset-v1:MITx+8.01.1x+3T2018+type@asset+block/images_k_w1_motion_1d_dnd_k_w1_motion_1d_dnd_dnd_label3.png", "label": "", "can_reuse": "", "id": "C"}], "one_per_target": "true", "label_bg_color": "rgb(222, 139, 238)", "targets": [{"w": "43", "id": "1", "h": "61", "x": "406", "y": "0"}, {"w": "43", "id": "2", "h": "61", "x": "406", "y": "80"}, {"w": "43", "id": "3", "h": "61", "x": "406", "y": "160"}], "base_image": "/assets/courseware/v1/fbe85d72177ef29dc0b2f61a8e774a60/asset-v1:MITx+8.01.1x+3T2018+type@asset+block/images_k_w1_motion_1d_dnd_k_w1_motion_1d_dnd_dnd.png"}</div>
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L2Q4-2: Graphical Representation of the Motion of Three Objects Part 2
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Match the correct function to the corresponding motion graph by dragging [mathjaxinline]x_ A(t)[/mathjaxinline], [mathjaxinline]x_ B(t)[/mathjaxinline] and [mathjaxinline]x_ C(t)[/mathjaxinline]. </p>
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L2Q4-3: Graphical Representation of the Motion of Three Objects Part 3
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Rank the x-components of the displacement from [mathjaxinline]t=2.5 \; s[/mathjaxinline] to [mathjaxinline]t=5\; s[/mathjaxinline] </p>
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle \Delta x\; =\; x\; (5\; s)\; -\; x\; (2.5\; s) \, .[/mathjaxinline]
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<td style="width:20%; border:none" class="eqnnum">&#160;</td>
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L2Q4-4: Graphical Representation of the Motion of Three Objects Part 4
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Rank the x-components of the displacement from [mathjaxinline]t=0 \; s[/mathjaxinline] to [mathjaxinline]t=10\; s[/mathjaxinline]. </p>
<table id="a0000000004" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000005">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Delta x\; =\; x\; (10\; s)\; -\; x\; (0\; s) \, .[/mathjaxinline]
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L2Q4-5: Graphical Representation of the Motion of Three Objects Part 5
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Rank the x-components of the average velocity from [mathjaxinline]t=5 \; s[/mathjaxinline] to [mathjaxinline]t=7.5\; s[/mathjaxinline] </p>
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L2Q4-6: Graphical Representation of the Motion of Three Objects Part 6
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Rank the x-components of the instantaneous velocity at [mathjaxinline]t=7.5 \; s[/mathjaxinline]. </p>
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For example, if you think the correct ranking is [mathjaxinline]v_{x,A} &lt; v_{x,B} &lt; v_{x,C}[/mathjaxinline], drag the terms to construct that statement. </p>
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L2Q4-7: Graphical Representation of the Motion of Three Objects Part 7
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Rank the x-components of the instantaneous acceleration at [mathjaxinline]t=5.5 \; s[/mathjaxinline]. </p>
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