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<h2 class="hd hd-2 unit-title">W1PSv1: Three Questions Before Starting</h2>
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This sequence will provide some guided problems and example problems. We start with a brief note about problem solving strategies.
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<h3 class="hd hd-2">W1PS1: Three Questions Before Starting</h3>
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<h4 class="hd hd-5">Transcripts</h4>
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<h2 class="hd hd-2 unit-title">W1PSQ1/W1PSv2: Category 1. Multiple Objects - one stage of motion each</h2>
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<p><b>Category 1. Two Objects, one stage of motion each.</b></p><p>There are two problems below</p>
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<h3 class="hd hd-3 problem-header" id="ps_week1ps1_1-problem-title" aria-describedby="block-v1:MITx+8.01.1x+3T2018+type@problem+block@ps_week1ps1_1-problem-progress" tabindex="-1">
W1PSQ1-1: Track Event (Practice)
</h3>
<div class="problem-progress" id="block-v1:MITx+8.01.1x+3T2018+type@problem+block@ps_week1ps1_1-problem-progress"></div>
<div class="problem">
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<p>
Runners A and B are nearing the end of a race. The last portion of track before the finish line is straight. Both runners are moving at a constant speed of [mathjaxinline]5[/mathjaxinline] m/s. When runner A is [mathjaxinline]50[/mathjaxinline] m behind runner B, she starts to accelerate at a constant rate of [mathjaxinline]0.25[/mathjaxinline] m/s[mathjaxinline]^2[/mathjaxinline]. At this moment, runner B, still going a constant speed, is [mathjaxinline]100[/mathjaxinline] m from the finish line. </p>
<p><b class="bfseries">(Part a)</b> How long (in seconds) does it take runner B to cross the finish line? <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps_week1ps1_1_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ps_week1ps1_1_2_1" id="input_ps_week1ps1_1_2_1" aria-describedby="trailing_text_ps_week1ps1_1_2_1 status_ps_week1ps1_1_2_1" value="" size="8"/>
<span class="trailing_text" id="trailing_text_ps_week1ps1_1_2_1">[mathjaxinline]\mathrm{s}[/mathjaxinline]</span>
<span class="status unanswered" id="status_ps_week1ps1_1_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_ps_week1ps1_1_2_1" class="answer"/>
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</div></div> </p>
<p><b class="bfseries">(Part a)</b> Who wins the race? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ps_week1ps1_1_3_1">
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<input type="radio" name="input_ps_week1ps1_1_3_1" id="input_ps_week1ps1_1_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ps_week1ps1_1_3_1-choice_1-label" for="input_ps_week1ps1_1_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ps_week1ps1_1_3_1"> <text> A</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_ps_week1ps1_1_3_1" id="input_ps_week1ps1_1_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="ps_week1ps1_1_3_1-choice_2-label" for="input_ps_week1ps1_1_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_ps_week1ps1_1_3_1"> <text> B</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_ps_week1ps1_1_3_1" id="input_ps_week1ps1_1_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="ps_week1ps1_1_3_1-choice_3-label" for="input_ps_week1ps1_1_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_ps_week1ps1_1_3_1"> <text> tie</text>
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<h3 class="hd hd-3 problem-header" id="ps_week1ps1_3-problem-title" aria-describedby="block-v1:MITx+8.01.1x+3T2018+type@problem+block@ps_week1ps1_3-problem-progress" tabindex="-1">
W1PSQ1-2: Shooting an apple (Practice)
</h3>
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</center>
<p>
An apple is suspended a height [mathjaxinline]h[/mathjaxinline] above the ground. A physics demo instructor has set up a projectile gun a horizontal distance [mathjaxinline]d[/mathjaxinline] away from the apple. The projectile is initially a height [mathjaxinline]s[/mathjaxinline] above the ground. The demo instructor fires the projectile with an initial velocity of magnitude [mathjaxinline]v_0[/mathjaxinline] just as the apple is released. Find [mathjaxinline]\theta _0[/mathjaxinline], the angle at which the projectile gun must be aimed in order for the projectile to strike the apple. Ignore air resistance. </p>
<p>
<p style="display:inline">[mathjaxinline]\theta _0=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps_week1ps1_3_2_1" class="text-input-dynamath capa_inputtype inline textline">
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</span>
<p id="answer_ps_week1ps1_3_2_1" class="answer"/>
<div id="display_ps_week1ps1_3_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps_week1ps1_3_2_1_dynamath" name="input_ps_week1ps1_3_2_1_dynamath"/>
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<h3 class="hd hd-2">W1PS2: Shooting the apple solution</h3>
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<h3 class="hd hd-3 problem-header" id="ps_week1ps1_2-problem-title" aria-describedby="block-v1:MITx+8.01.1x+3T2018+type@problem+block@ps_week1ps1_2-problem-progress" tabindex="-1">
W1PSQ1-3: Shooting an apple continued (Practice)
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<p>
Now assume that the bullet is fired at a time [mathjaxinline]t_0[/mathjaxinline] after the apple starts falling from the tree. Consider time [mathjaxinline]t = 0[/mathjaxinline] to be the instant of time when the bullet is fired, and [mathjaxinline](x=0,y=0)[/mathjaxinline], the origin of the coordinate system, to be on the ground at a distance [mathjaxinline]s[/mathjaxinline] below the location of the gun. Take [mathjaxinline]+x[/mathjaxinline] to be to the right and [mathjaxinline]+y[/mathjaxinline] to be upwards. The initial speed of the bullet is [mathjaxinline]v_0[/mathjaxinline] and the angle between the initial velocity and the horizontal is [mathjaxinline]\theta _0[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> Draw a qualitative sketch of [mathjaxinline]y_ a(t)[/mathjaxinline] and [mathjaxinline]y_ b(t)[/mathjaxinline], the [mathjaxinline]y[/mathjaxinline]-component of the apple's and bullet's positions as a function of time. </p>
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<span id="solution_ps_week1ps1_2_solution_1"/>
</div></p>
<p><b class="bfseries">(Part b)</b> Write the mathematical expression of [mathjaxinline]y_ b(t)[/mathjaxinline], the [mathjaxinline]y[/mathjaxinline]-component of the bullet's position as a function of time. Express your answer in terms of [mathjaxinline]t[/mathjaxinline], [mathjaxinline]s[/mathjaxinline], [mathjaxinline]h[/mathjaxinline], [mathjaxinline]d[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], v_0 for [mathjaxinline]v_0[/mathjaxinline], t_0 for [mathjaxinline]t_0[/mathjaxinline], and theta_0 for [mathjaxinline]\theta _0[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]y_{b}(t) =[/mathjaxinline]</p>
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<p><b class="bfseries">(Part c)</b> Write the mathematical expression for [mathjaxinline]y_ a(t)[/mathjaxinline], the [mathjaxinline]y[/mathjaxinline]-component of the apple's position as a function of time. Express your answer in terms of [mathjaxinline]t[/mathjaxinline], [mathjaxinline]s[/mathjaxinline], [mathjaxinline]h[/mathjaxinline], [mathjaxinline]d[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], v_0 for [mathjaxinline]v_0[/mathjaxinline], t_0 for [mathjaxinline]t_0[/mathjaxinline], and theta_0 for [mathjaxinline]\theta _0[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]y_{a}(t) =[/mathjaxinline]</p>
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<p><b class="bfseries">(Part d)</b> Write the mathematical expression of [mathjaxinline]x_ b(t)[/mathjaxinline], the x-component of the bullet's position as a function of time. Express your answer your answer in terms of [mathjaxinline]t[/mathjaxinline], [mathjaxinline]s[/mathjaxinline], [mathjaxinline]h[/mathjaxinline], [mathjaxinline]d[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], v_0 for [mathjaxinline]v_0[/mathjaxinline], t_0 for [mathjaxinline]t_0[/mathjaxinline], and theta_0 for [mathjaxinline]\theta _0[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]x_{b}(t) =[/mathjaxinline]</p>
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<div id="display_ps_week1ps1_2_4_1" class="equation">`{::}`</div>
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<span id="solution_ps_week1ps1_2_solution_4"/>
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<p><b class="bfseries">(Part e)</b> Assume that [mathjaxinline]v_0=50[/mathjaxinline] m/s, [mathjaxinline]h=41[/mathjaxinline] m, [mathjaxinline]s=1[/mathjaxinline] m, [mathjaxinline]t_0=1[/mathjaxinline] s, [mathjaxinline]\theta _0 = 20^ o[/mathjaxinline], and [mathjaxinline]g\approx 10[/mathjaxinline] m/s[mathjaxinline]^2[/mathjaxinline]. Calculate [mathjaxinline]t_ h[/mathjaxinline], the instant of time when the bullet hits the apple, and [mathjaxinline]d[/mathjaxinline], the distance between the gun and the tree. Express your answer in seconds for [mathjaxinline]t_ h[/mathjaxinline], and in meters for [mathjaxinline]d[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]t_ h =[/mathjaxinline] </p>
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<span class="trailing_text" id="trailing_text_ps_week1ps1_2_5_1">[mathjaxinline]\mathrm{s}[/mathjaxinline]</span>
<span class="status unanswered" id="status_ps_week1ps1_2_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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</p>
<p>
<p style="display:inline">[mathjaxinline]d =[/mathjaxinline] </p>
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<div class="unanswered inline">
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<span class="trailing_text" id="trailing_text_ps_week1ps1_2_6_1">[mathjaxinline]\mathrm{m}[/mathjaxinline]</span>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<h2 class="hd hd-2 unit-title">W1PSQ2/W1PSv3: Category 2: Multiple Stages of Motion</h2>
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<p><b> Category 2: Multiple Stages of Motion.</b></p><p>There are two problems here. The second problem has a help video that shows how to set up the problem.</p>
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W1PSQ2-1: Accelerating Bike (Practice)
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At [mathjaxinline]t=0[/mathjaxinline], a bike starting at rest at [mathjaxinline]x=0[/mathjaxinline] accelerates in the +x-direction with an acceleration given by </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]a(t) = \left\{ \begin{array}{ll} At-Bt^{3} &amp; \quad 0 \leq t \leq t_1 \\ 0 &amp; \quad t &gt; t_1 \end{array} \right.[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
where [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]B[/mathjaxinline] are positive constants in SI units, and [mathjaxinline]t_1 = (A/B)^{1/2}[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> What are the units of the constant [mathjaxinline]A[/mathjaxinline]? Enter m for meters and s for seconds. </p>
<p>
<p style="display:inline">Units of the constant A: </p>
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<p><b class="bfseries">(Part b)</b> What are the units of the constant [mathjaxinline]B[/mathjaxinline]? Enter m for meters and s for seconds. </p>
<p>
<p style="display:inline">Units of the constant B: </p>
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<p><b class="bfseries">(Part c)</b> What is the velocity of the bike as a function of time for times [mathjaxinline]0\lt t \lt t_{1}[/mathjaxinline]? </p>
<p>
<p style="display:inline">[mathjaxinline]v(t) =[/mathjaxinline] </p>
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<span id="solution_ps_week1ps2_1_solution_2"/>
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<p><b class="bfseries">(Part d)</b> Find the bike's velocity as a function of time for times [mathjaxinline]t \gt t_{1}[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]v(t) =[/mathjaxinline] </p>
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<span id="solution_ps_week1ps2_1_solution_3"/>
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<p><b class="bfseries">(Part e.)</b> Graph [mathjaxinline]x(t)[/mathjaxinline] for times [mathjaxinline]t &gt; 0[/mathjaxinline]. <div class="solution-span">
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W1PSQ2-2: Braking Car (Practice)
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<p>
At time [mathjaxinline]t=0[/mathjaxinline], a car moving along the +[mathjaxinline]x[/mathjaxinline]-axis passes through [mathjaxinline]x=0[/mathjaxinline] with a constant velocity of magnitude [mathjaxinline]v_0[/mathjaxinline]. At some time later, [mathjaxinline]t_1[/mathjaxinline], it starts to slow down. The acceleration of the car as a function of time is given by: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]a(t) = \left\{ \begin{array}{ll} 0 &amp; \quad 0 \leq t \leq t_1 \\ -c(t-t_1) &amp; \quad t_1 \lt t \leq t_2 \end{array} \right.[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
where [mathjaxinline]c[/mathjaxinline] is a positive constants in SI units, and [mathjaxinline]t_1 \lt t \leq t_2[/mathjaxinline] is the given time interval for which the car is slowing down. The goal of the problem is to find the car's position as a function of time between [mathjaxinline]t_{1} \lt t \lt t_2[/mathjaxinline]. Express your answer in terms of v_0 for [mathjaxinline]v_0[/mathjaxinline], t_1 for [mathjaxinline]t_1[/mathjaxinline], t_2 for [mathjaxinline]t_2[/mathjaxinline], and [mathjaxinline]c[/mathjaxinline] as needed. </p>
<p><b class="bfseries">(Part a)</b>. What is [mathjaxinline]v(t)[/mathjaxinline], the velocity of the car as a function of time during the time interval [mathjaxinline]0 \leq t \leq t_1[/mathjaxinline]? <p style="display:inline">[mathjaxinline]v(t) =[/mathjaxinline] </p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps_week1ps2_2_2_1" class="text-input-dynamath capa_inputtype inline textline">
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</div></div></p>
<p><div class="solution-span">
<span id="solution_ps_week1ps2_2_solution_1"/>
</div><b class="bfseries">(Part b)</b>. What is [mathjaxinline]x(t)[/mathjaxinline], the position of the car as a function of time during the time interval [mathjaxinline]0 \leq t \leq t_1[/mathjaxinline]? <p style="display:inline">[mathjaxinline]x(t) =[/mathjaxinline] </p><div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps_week1ps2_2_3_1" class="text-input-dynamath capa_inputtype inline textline">
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<span id="solution_ps_week1ps2_2_solution_2"/>
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<p><b class="bfseries">(Part c)</b>. What is [mathjaxinline]v(t)[/mathjaxinline], the velocity of the car as a function of time during the time interval [mathjaxinline]t_1 \lt t \leq t_2[/mathjaxinline]? </p>
<p>
<p style="display:inline">[mathjaxinline]v(t) =[/mathjaxinline] </p>
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<span id="solution_ps_week1ps2_2_solution_3"/>
</div></p>
<p><b class="bfseries">(Part d)</b>. What is [mathjaxinline]x(t)[/mathjaxinline], the position of the car as a function of time during the time interval [mathjaxinline]t_1 \lt t \leq t_2[/mathjaxinline]? </p>
<p>
<p style="display:inline">[mathjaxinline]x(t) =[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">W1PSQ3/W1PSv4: Category 3: Special Conditions to Apply</h2>
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<p><b>Category 3: Special Conditions on the motion.</b></p>
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W1SPQ3-1: Particle moving between plates - part 1 (Practice)
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A particle of mass [mathjaxinline]m[/mathjaxinline] is moving along the +[mathjaxinline]x[/mathjaxinline]-axis with a constant speed [mathjaxinline]v_0[/mathjaxinline]. At time [mathjaxinline]t = 0[/mathjaxinline] it enters a region in space between two parallel plates which are separated by a distance [mathjaxinline]h[/mathjaxinline] and are contained in the [mathjaxinline]x[/mathjaxinline]-[mathjaxinline]z[/mathjaxinline] plane. The figure shows a side view of the plates, the +[mathjaxinline]z[/mathjaxinline] axis is out of the screen. While the particle is between the plates, it is acted on by both gravity and by a time varying force that points upward (along the +[mathjaxinline]y[/mathjaxinline]-direction) and has magnitude [mathjaxinline]F = bt[/mathjaxinline], where [mathjaxinline]b[/mathjaxinline] is a positive constant that is sufficiently large such that the particle hits the top plate without ever touching the bottom plate. </p>
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<p><b class="bfseries">(Part 1)</b> What are the units of the constant [mathjaxinline]b[/mathjaxinline]? Enter m for meters, s for seconds and kg for kilograms. </p>
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<p style="display:inline">Units of the constant [mathjaxinline]b[/mathjaxinline]: </p>
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Make a sketch of what you think the trajectory of the particle is as it moves through the plates. Draw a coordinate system showing the position of the particle at time [mathjaxinline]t[/mathjaxinline]. Clearly indicate your origin, choice of axis, and draw in the coordinate functions for the position of the particle at time [mathjaxinline]t[/mathjaxinline]? </p>
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W1PSQ3-2: Particle moving between plates - part 2 (Practice)
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<p><b class="bfseries">(Part 2)</b> Calculate [mathjaxinline]t_1[/mathjaxinline], the time it takes the particle to reach the lowest point of its trajectory. For that purpose, draw the force free body diagram of the particle and apply Newton's 2nd law to the particle to obtain the particle's acceleration. Use the kinematics equation to calculate [mathjaxinline]t_1[/mathjaxinline]. Express your answer in terms of some or all of the given variables [mathjaxinline]m[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], [mathjaxinline]h[/mathjaxinline], [mathjaxinline]b[/mathjaxinline], and v_0 for [mathjaxinline]v_0[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]t_1 =[/mathjaxinline] </p>
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<h3 class="hd hd-3 problem-header" id="ps_week1ps3_2c-problem-title" aria-describedby="block-v1:MITx+8.01.1x+3T2018+type@problem+block@ps_week1ps3_2c-problem-progress" tabindex="-1">
W1PSQ3-3: Particle moving between plates - part 3 (Practice)
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<p><b class="bfseries">(Part 3)</b> What is the minimum possible value of [mathjaxinline]b[/mathjaxinline] such that the particle does not hit the lower plate? Express your answer in term sof some or all of the given variables [mathjaxinline]m[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], [mathjaxinline]h[/mathjaxinline], [mathjaxinline]b[/mathjaxinline], and v_0 for [mathjaxinline]v_0[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]b =[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">W1PSQ4/W1PSv5: Combining the 3 Categories</h2>
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Here is an example problem that combines the 3 categories. Two objects, one has two
stages of motion, the other only one stage.
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<h3 class="hd hd-3 problem-header" id="ps_week1ps4-problem-title" aria-describedby="block-v1:MITx+8.01.1x+3T2018+type@problem+block@ps_week1ps4-problem-progress" tabindex="-1">
W1PSQ4: A car and a bicycle (Practice)
</h3>
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<p>
You are in a car standing by a traffic light and at time [mathjaxinline]t=0[/mathjaxinline] the light turns green. You start to accelerate during the first [mathjaxinline]t_1[/mathjaxinline] seconds so that the acceleration of your car is given by: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]a_1(t) = \left\{ \begin{array}{ll} b_1 &amp; \quad 0 \leq t \leq t_1 \\ 0 &amp; \quad t_1 &lt; t \leq t_2 \end{array} \right.[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
where [mathjaxinline]b_1[/mathjaxinline] is a positive constant. </p>
<p>
At the instant the light turns green a cyclist passes through the intersection moving with a speed [mathjaxinline]v_0[/mathjaxinline] in the same direction as your car is moving. At that instant, the cyclist starts to brake with a constant acceleration of magnitude [mathjaxinline]b_2[/mathjaxinline]. At time [mathjaxinline]t = t_2[/mathjaxinline] the cyclist stops at the same location where you are. </p>
<p>
The goal of the problem is to calculate the value of [mathjaxinline]b_2[/mathjaxinline] in terms of the given variables, [mathjaxinline]b_1[/mathjaxinline], [mathjaxinline]v_0[/mathjaxinline], [mathjaxinline]t_1[/mathjaxinline] and [mathjaxinline]t_2[/mathjaxinline]. </p>
<p><b class="bfseries">First:</b> Describe the motion of the car. Given the car's acceleration [mathjaxinline]a_1(t)[/mathjaxinline], find its velocity and its position as a function of time: </p>
<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]v_1(t)[/mathjaxinline] , the car's velocity as a funcion of time. Express your answer in terms of [mathjaxinline]t[/mathjaxinline], t_1 for [mathjaxinline]t_1[/mathjaxinline], and b_1 for [mathjaxinline]b_1[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">For times [mathjaxinline]0 \leq t \leq t_1[/mathjaxinline], [mathjaxinline]v_1(t) =[/mathjaxinline] </p>
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<p style="display:inline">For times [mathjaxinline]t_1 &lt; t \leq t_2[/mathjaxinline], [mathjaxinline]v_1(t) =[/mathjaxinline]</p>
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</p>
<p><b class="bfseries">(Part b)</b> Calculate [mathjaxinline]x_1(t)[/mathjaxinline] , the car's position as a funcion of time. Express your answer in terms of [mathjaxinline]t[/mathjaxinline], t_1 for [mathjaxinline]t_1[/mathjaxinline], and b_1 for [mathjaxinline]b_1[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">For times [mathjaxinline]0 \leq t \leq t_1[/mathjaxinline], [mathjaxinline]x_1(t) =[/mathjaxinline] </p>
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</p>
<p>
<p style="display:inline">For times [mathjaxinline]t_1 &lt; t \leq t_2[/mathjaxinline], [mathjaxinline]x_1(t) =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_ps_week1ps4_5_1" class="text-input-dynamath capa_inputtype inline textline">
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</p>
<p><b class="bfseries">Second:</b> Describe the motion of the bicycle. Given the bicycle's acceleration [mathjaxinline]a_2(t)[/mathjaxinline], find its velocity and its position: </p>
<p><b class="bfseries">(Part c)</b> Calculate [mathjaxinline]v_2(t)[/mathjaxinline] , the bicycle's velocity as a funcion of time. Express your answer in terms of [mathjaxinline]t[/mathjaxinline], b_2 for [mathjaxinline]b_2[/mathjaxinline], and v_0 for [mathjaxinline]v_0[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">For times [mathjaxinline]0 \leq t \leq t_2[/mathjaxinline], [mathjaxinline]v_2(t) =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_ps_week1ps4_6_1" class="text-input-dynamath capa_inputtype inline textline">
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<span class="status unanswered" id="status_ps_week1ps4_6_1" data-tooltip="Not yet answered.">
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</p>
<p><b class="bfseries">(Part d)</b> Calculate [mathjaxinline]x_2(t)[/mathjaxinline] , the bicycle's position as a funcion of time. Express your answer in terms of [mathjaxinline]t[/mathjaxinline], b_2 for [mathjaxinline]b_2[/mathjaxinline], and v_0 for [mathjaxinline]v_0[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">For times [mathjaxinline]0 \leq t \leq t_2[/mathjaxinline], [mathjaxinline]x_2(t) =[/mathjaxinline] </p>
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<p><b class="bfseries">Third:</b> Find the value of [mathjaxinline]b_2[/mathjaxinline]. We know that the bicycle stops at [mathjaxinline]t=t_2[/mathjaxinline], this condition is expressed as: </p>
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[mathjaxinline]v_2(t_2) = 0[/mathjaxinline] (eq. 1) </p>
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We also know that the bicycle and the car are at the same location when the bicycle stops. This condition implies: </p>
<p>
[mathjaxinline]x_2(t_2) = x_1(t_2)[/mathjaxinline] (eq. 2) </p>
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Use (eq.1) and (eq. 2) to obtain the value of [mathjaxinline]b_2[/mathjaxinline]. Express your answer in terms of t_1 for [mathjaxinline]t_1[/mathjaxinline], b_1 for [mathjaxinline]b_1[/mathjaxinline], and v_0 for [mathjaxinline]v_0[/mathjaxinline]. Do not use [mathjaxinline]t_2[/mathjaxinline] in your answer. </p>
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<p style="display:inline">[mathjaxinline]b_2[/mathjaxinline]= </p>
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