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<h2 class="hd hd-2 unit-title">Topics and Reading List</h2>
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<p> In this lesson we will start our mathematical description of One Dimensional (1D) Kinematics. 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> Position and Displacement: <a href="/courses/course-v1:MITx+8.01.1x+3T2018/pdfbook/0/chapter/4/3"><i class="it">chapter 4, section 2</i></a></li><li> Velocity: <a href="/courses/course-v1:MITx+8.01.1x+3T2018/pdfbook/0/chapter/4/4"><i class="it">chapter 4, section 3</i></a></li></ul>
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<h2 class="hd hd-2 unit-title">L1v1: Coordinate Systems and Unit Vectors in 1D</h2>
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<h3 class="hd hd-2">L1v1: Coordinate Systems and Unit Vectors in 1D</h3>
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<h2 class="hd hd-2 unit-title">L1v2: Position Vector in 1D</h2>
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<h2 class="hd hd-2 unit-title">L1v3: Displacement Vector in 1D</h2>
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<p><b>Distance:</b> Is the length of the path travelled by an object between two points
in space. From its definition, the distance is a scalar and it is always a positive quantity.
</p><p><b> Displacement:</b> Is the change in the position of an object. If at time \(t=t_1\) the object
is at position \(\vec{r}(t_1)\), and at a later time \(t=t_2 > t_1\) the object is at position
\(\vec{r}(t_2)\), the displacement vector is defined as \(\Delta \vec{r} = \vec{r}(t_2) - \vec{r}(t_1)\).
In one dimension, the displacement vector has one component. For example, if the motion is along the x-axis,
the displacement vector becomes \(\Delta \vec{r} = \Delta x \hat{i} = (x(t_2) - x(t_1))\hat{i}\).
The component of the displacement vector can be positive, when the final position is larger than the initial
one. It can be negative, when the final position is smaller than the initial one. It can aslo be zero, if
the object ends at the starting point.
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<h2 class="hd hd-2 unit-title">L1Q1: Displacement vs. Distance, 1D</h2>
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L1Q1: Displacement vs. Distance, 1D
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Peter is training for a race. He first sprints \(24 \,\text{m}\) north. He then walks due south three-quarters of the way back to his starting position.
</p>
<p>In the following, take north to be the \(+x\) direction.</p>
<table>
<tr>
<td style="padding-top:2em;vertical-align:top;padding-right:4em">
<strong>Displacement</strong>
</td>
<td style="padding-top:2em;vertical-align:top">
<strong>Distance</strong>
</td>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><span>What is Peter's displacement during the sprint?<br/></span>
<span> \(\Delta x_{\text{sprint}}=\) </span>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><span>Through what distance does Peter travel during the sprint?<br/></span>
<span> \(d_{\text{sprint}}=\) </span>
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<span> \(\Delta x_{\text{walk}}=\) </span>
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<span class="trailing_text" id="trailing_text_lore_ls01_19707_47dafc126dfe4e3bacb9fe55271f5d05_4_1">m</span>
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<span> \(d_{\text{walk}}=\) </span>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><span>What is Peter's total displacement?<br/></span>
<span> \(\Delta x_{\text{total}}=\) </span>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 6" role="group"><span>What is the total distance through which Peter travels?<br/></span>
<span> \(d_{\text{total}}=\) </span>
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<span class="trailing_text" id="trailing_text_lore_ls01_19707_47dafc126dfe4e3bacb9fe55271f5d05_7_1">m</span>
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<h3 class="hd hd-2">L1v4: Average Velocity in 1D</h3>
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<h2 class="hd hd-2 unit-title">L1Q2: Average Velocity and Average Speed</h2>
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L1Q2-1: Average Velocity vs. Average Speed - Part I
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<p><b class="bfseries">Note:</b> In this question and in the following problem the average speed and the average velocity, are defined and compared. Although, the concept of average speed is frequently used in high school and in everyday life, in this course we will use average velocity. </p>
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Consider an object moving along the x-axis. During a given time interval [mathjaxinline]\Delta t[/mathjaxinline] the object has covered a distance [mathjaxinline]d[/mathjaxinline] and its displacement is given by [mathjaxinline]\Delta x[/mathjaxinline]. </p>
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Drag and drop the proper quantities to define the average velocity and the average speed: </p>
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L1Q2-2: Average Velocity vs. Average Speed - Part II
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<p>
Peter is training for a race. He sprints \(24 \,\text{m}\) north in \(4 \,\text{s}\), then walks due south back to his starting position in a time \(12 \,\text{s}\).
</p>
<p>In the following, take north as the \(+x\) direction.</p>
<table>
<tr>
<td style="padding-top:2em;vertical-align:top;padding-right:4em">
<strong>Average Velocity</strong>
</td>
<td style="padding-top:2em;vertical-align:top">
<strong>Average Speed</strong>
</td>
</tr>
<tr>
<td style="padding-top:2em;vertical-align:top;padding-right:4em">
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><span>What is Peter's <b>average velocity</b> as he sprints?</span>
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<span class="trailing_text" id="trailing_text_lore_ls01_19737_de5088fc92c74394a64939d1adb72ae9_2_1">[mathjaxinline]\hat{i}[/mathjaxinline] m/s</span>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><span>What is Peter's <b>average speed</b> as he sprints?</span>
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<span class="trailing_text" id="trailing_text_lore_ls01_19737_de5088fc92c74394a64939d1adb72ae9_3_1">m/s</span>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><span>What is Peter's <b>average velocity</b> as he walks?</span>
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<span class="trailing_text" id="trailing_text_lore_ls01_19737_de5088fc92c74394a64939d1adb72ae9_4_1">[mathjaxinline]\hat{i}[/mathjaxinline] m/s</span>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><span>What is Peter's <b>average speed</b> as he walks?</span>
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<span class="trailing_text" id="trailing_text_lore_ls01_19737_de5088fc92c74394a64939d1adb72ae9_6_1">[mathjaxinline]\hat{i}[/mathjaxinline] m/s</span>
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<h2 class="hd hd-2 unit-title">L1Q3: Instantaneous Velocity</h2>
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L1Q3: Instantaneous Velocity
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The graph shows the position as a function of time for two trains running on parallel tracks. For times greater than [mathjaxinline]t = 0[/mathjaxinline], which of the following is true: </p>
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<text> a. At time [mathjaxinline]t_ B[/mathjaxinline], both trains have the same instantaneous velocity.</text>
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<text> b. The speed of both trains increases all the time.</text>
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<text> c. Both trains have the same instantaneous velocity at some time before [mathjaxinline]t_ B[/mathjaxinline].</text>
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<text> d. Somewhere on the graph, the velocities of both trains are changing in the same way.</text>
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<h2 class="hd hd-2 unit-title">Derivatives</h2>
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<p><b> List of useful derivatives: </b></p><ul><li><p>Derivative of a polynomial function:</p><p>If \(\displaystyle x(t)=At^n \Longrightarrow \frac{dx}{dt}=nAt^{n-1} \)</p><p>where \(A\) and \(n\) are constants.</p></li><li><p>Derivative of an exponential function:</p><p>If \(\displaystyle x(t)=A e^{bt} \Longrightarrow \frac{dx}{dt}=Ab e^{bt} \)</p><p>where \(A\) and \(b\) are constants.</p></li><li><p>Derivative of a logarithmic function:</p><p>If \(\displaystyle x(t)=A\ln(b+ct) \Longrightarrow \frac{dx}{dt}=\frac{Ac}{b+ct} \)</p><p>where \(A\), \(b\) and \(c\) are constants.</p></li><li><p>Derivative of sine:</p><p>If \( \displaystyle x(t)=A\sin(b+ct) \Longrightarrow \frac{dx}{dt}=Ac \cos(b+ct) \)</p><p>where \(A\), \(b\) and \(c\) are constants.</p></li><li><p>Derivative of cosine:</p><p>If \( \displaystyle x(t)=A\cos(b+ct) \Longrightarrow \frac{dx}{dt}=-Ac \sin(b+ct) \)</p><p>where \(A\), \(b\) and \(c\) are constants.</p></li></ul><p><b>External References </b></p><p><a href="http://web.mit.edu/wwmath/calculus/differentiation/polynomials.html" target="_blank">1. World Web Math</a></p><p><a href="http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/video-lectures/lecture-3-derivatives/" target="_blank">2. OCW: Single Variable Calculus - Video Lecture 3</a></p><p><a href="http://www.wolframalpha.com/" target="_blank">3. Wolfram Alpha </a></p>
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<h2 class="hd hd-2 unit-title">L1Q4: Instantaneous Velocity from Position</h2>
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L1Q4: Instantaneous Velocity from Position
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<p>A ship is moving in a straight line along the x-axis. At time, \(t = 0\), the ship is at the origin. At a
later time, \(t\), the ship's \(x\)-component of position in meters is given by \(x(t) = a t^4 - bt^3\).</p>
<p><b>(Part a) </b> What is the \(x\)-component of the ship's instantaneous velocity at time \(t\)?</p>
<p>
<p style="display:inline">[mathjaxinline] v_x(t) =[/mathjaxinline] </p>
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<p><b>(Part b)</b> If \(a = 0.5 \) m/s\(^{4}\) and \(b = 0.25 \) m/s\(^{3}\),
what is the \(x\)-component of the ship's instantaneous velocity at time \(t=2.5 \,\text{s}\)?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><p style="display:inline">\(v=\) </p>
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<p><b>(Part c)</b> If \(a = 0.5 \) m/s\(^{4}\) and \(b = 0.25 \) m/s\(^{3}\), what is the \(x\)-component of the ship's average velocity between \(t=2 \,\text{s}\) and \(t=3 \,\text{s}\)? </p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><p style="display:inline"> \(v_{\text{avg,x}} = \)</p>
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<p><b>(Part d)</b> If \(a = 0.5 \) m/s\(^{4}\) and \(b = 0.25 \) m/s\(^{3}\), what is the \(x\)-component of the ship's average velocity between \(t=2.55 \,\text{s}\) and \(t=2.65 \,\text{s}\)?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><p style="display:inline">\(v_{\text{avg,x}} = \)</p>
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<span class="trailing_text" id="trailing_text_lore_ls01_19749_b4caaa0e49b8467db75049eba62ed7c8_5_1">m/s</span>
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<h2 class="hd hd-2 unit-title">L1WE1/L1v6: Worked Example - Derivatives in Kinematics</h2>
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<p>This is a Solved Example problem using derivatives. The solution is given in the video below. The question is not worth any credit, but is here for you to read before watching the video and for you to test yourself with if desired. </p>
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L1WE1: Model Rocket (Worked Example)
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A home-built model rocket is launched straight up into the air. At time [mathjaxinline]t = 0[/mathjaxinline] the rocket is at rest, about to be launched. The position of the rocket as a function of time is given by: </p>
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[mathjaxinline]y(t) = \frac{1}{2}(a_0-g)t^2 - \frac{1}{30}\frac{a_0 }{t_{0}^{4}}t^{6}[/mathjaxinline] for [mathjaxinline]0 \lt t \lt t_0[/mathjaxinline] </p>
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where [mathjaxinline]a_0[/mathjaxinline] is a positive constant, [mathjaxinline]g[/mathjaxinline] is the acceleration of gravity and [mathjaxinline]a_0 &gt; g[/mathjaxinline]. The contant [mathjaxinline]t_0[/mathjaxinline] is the amount of time that the fuel takes to burn out. Express your answer in terms of g, t, a_0 for [mathjaxinline]a_0[/mathjaxinline], and t_0 for [mathjaxinline]t_0[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> Find [mathjaxinline]a[/mathjaxinline], the [mathjaxinline]y[/mathjaxinline]-component of the acceleration as a function of time. </p>
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<p style="display:inline">[mathjaxinline]a_{y} =[/mathjaxinline] </p>
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<h3 class="hd hd-2">L1v6: Worked Example: Derivatives in Kinematics</h3>
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