\hfill \hfill } %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \geometry{left=0.5in,right=0.5in,top=0.5in,bottom=0.4in} \pagestyle{fancy} \lhead{\color{blue} \Large Lecture notes} \chead{\color{black} \LARGE Average Velocity - Part 2} \rhead{\large page \ \thepage} \cfoot{} \lfoot{\small \copyright $\;$ Hidegkuti, 2014} \rfoot{\small Last revised: September 8, 2014} \textwidth 7.5in \textheight 9.6in \setlength{\headheight}{29pt} \setlength{\parindent}{0in} \begin{document} In what follows, we will consider the motion of an object that is moving along a vertical line. \ (Imagine an elevator in a very tall building that also have lots of underground floors.) \ We will describe the vertical position of the object by $L\left( t\right) $, a location function. \ \ $t$ will denote time, measured in seconds, and $L$ will denote the vertical position, measured in meters. \ So, $L\left( 5\right) =-3$ means that $5$ seconds after we start monitoring the object, it is $3$ meters below ground level. \ $L\left( 10\right) =2$. \bigskip \begin{enumerate} \item Suppose the location function of an object is given by $L\left( t\right) =-3t+20$. \ a) \ Where is the object at the start (i.e. when we start monitoring its motion)? b) \ Where is the object $4$ seconds after we start monitoring its motion? c) \ Where is the object $7$ seconds after we start monitoring its motion? \item Suppose the location function of an object is given by $L\left( t\right) =-t^{2}+4t+8$. \ a) \ Where is the object at the start (i.e. when we start monitoring its motion)? b) \ Where is the object $3$ seconds after we start monitoring its motion? c) \ Where is the object $7$ seconds after we start monitoring its motion? \end{enumerate} The \textbf{displacement} of an object is expressing the change in its location. \ If $L\left( t_{1}\right) =13$ and later, $L\left( t_{2}\right) =21$, then between $t_{1}$ and $t_{2}$, the object moved from a height of $% 13 $ meters to a height of $21$ meters. \ The displacement (often denoted by $s$) is the change that has occured: \ \begin{equation*} s=L\left( t_{2}\right) -L\left( t_{1}\right) =21\unit{m}-13\unit{m}=8\unit{m} \end{equation*}% Another notation for displacement is using the capital Greek letter $\Delta $ to express change. \ Since displacement is the change in location, it can also be denoted by $\Delta L$.\bigskip The displacement can easily be negative. \ Imagine if an object is moving downward. \ If $L\left( t_{1}\right) =13$ and later, $L\left( t_{2}\right) =2 $, then the displacement is% \begin{equation*} s=L\left( t_{2}\right) -L\left( t_{1}\right) =2\unit{m}-13\unit{m}=-11\unit{m% } \end{equation*}% A negative displacement indicates that the object has moved downward between $t_{1}$ and $t_{2}$. \bigskip \begin{enumerate} \item[3.] Suppose the location function of an object is given by $L\left( t\right) =-3t+20$. a) \ Find the displacement that occurs during the first $5$ seconds. b) \ Find the displacement between $t_{1}=3\unit{s}$ and $t_{2}=7\unit{s}$. \ ($\unit{s}$ denotes seconds) \ \ \item[4.] Suppose the location function of an object is given by $L\left( t\right) =-t^{2}+4t+8$. \ a) \ Find the displacement that occurs during the first $3$ seconds. b) \ Find the displacement that occurs during the first $4$ seconds c) \ Find the displacement between $t_{1}=1\unit{s}$ and $t_{2}=5\unit{s}$. d) \ Find the displacement between $t_{1}=2\unit{s}$ and $t_{2}=6\unit{s}$. e) \ Find the displacement between $t_{1}=3\unit{s}$ and $t_{2}=7\unit{s}$% .\bigskip \pagebreak \end{enumerate} The \textbf{average velocity} of an object is defined as the displacement divided by the time it took to travel that much. \bigskip \begin{enumerate} \item[5.] Suppose the location function of an object is given by $L\left( t\right) =-3t+20$, where $t$ is measured in seconds and $L$ in meters. a) \ Compute the average velocity of the object between $t_{1}=4\unit{s}$ and $t_{2}=10\unit{s}$. \ b) \ Compute the average velocity between $t_{1}=5\unit{s}$ and $t_{2}=8% \unit{s}$. \item[6.] Suppose the location function of an object is given by $L\left( t\right) =-t^{2}+4t+8$, where $t$ is measured in seconds and $L$ in meters. a) \ \ Compute the average velocity between $t_{1}=0\unit{s}$ and $t_{2}=3% \unit{s}$. b) \ \ Compute the average velocity between $t_{1}=0\unit{s}$ and $t_{2}=4% \unit{s}$. c) \ \ Compute the average velocity between $t_{1}=5\unit{s}$ and $t_{2}=9% \unit{s}$. \ \ \ \bigskip \item[7.] Suppose that a small object is moving up and down along a vertical line. \ We monitor the location of the object as a function of time. \ We set ground level to represent a height (or vertical location) to be zero. \ In each case, graph the location function given the data on the height. $t$ is time, measured by seconds and $h$ is the height, measured in meters. \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|} \hline $t$ & $0$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ & $11$ & $12$ & $13$ & $14$ \\ \hline $h$ & $7$ & $5.25$ & $4$ & $3.25$ & $3$ & $3.25$ & $4$ & $5.25$ & $7$ & $% 9.25 $ & $12$ & $15.25$ & $19$ & $23.25$ & $28$ \\ \hline \end{tabular}% \bigskip a) \ Create a coordinate system to graph this data. \ \ Label both axis and set up a consistent scale on both of them. \ Then graph the data given. b) \ When is the object moving upward? c) \ What is the average velocity of the object between $t=0$ and $t=3$ seconds? d) \ What is the average velocity of the object between $t=5$ seconds and $% t=10$ seconds? \end{enumerate} \pagebreak \begin{center} {\LARGE Answers} \end{center} \begin{enumerate} \item a) \ \ $L\left( 0\right) =20\qquad $b) \ $L\left( 4\right) =8\qquad $% c) \ $L\left( 7\right) =-1$ \item a) \ $L\left( 0\right) =8\qquad $b) \ $L\left( 3\right) =11\qquad $c) \ $L\left( 7\right) =-13$ \item[3.] a) $-15\unit{m}\qquad $b) \ $-12\unit{m}$ \item[4.] a) \ $3\qquad $b) \ $0\qquad $c) \ $-8\qquad $d) \ $-16\qquad $e) \ $-24$ \item[5.] a) \ $-3\dfrac{\unit{m}}{\unit{s}}$\qquad b) \ $-3\dfrac{\unit{m}}{% \unit{s}}$ \item[6.] a) \ $1\dfrac{\unit{m}}{\unit{s}}\qquad $b) \ $0\dfrac{\unit{m}}{% \unit{s}}\qquad $c) \ \ $-10\dfrac{\unit{m}}{\unit{s}}$ \item[7.] a) \ see below\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ b) \ \ after $% t=4 $ \ \ \ \ \ \ \ \ \ \ c) \ $-\dfrac{5}{4}\dfrac{\unit{ft}}{\unit{s}}$ \ \ \ \ \ d) \ d) \ $\ \dfrac{7}{4}\dfrac{\unit{ft}}{\unit{s}}$ \FRAME{itbpFX}{3.0113in}{1.7063in}{0in}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.0113in;height 1.7063in;depth 0in;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "0";xmax "15";xviewmin "0";xviewmax "15";yviewmin "-1";yviewmax "40";viewset"XY";rangeset"X";plottype 4;plotticks 1;num-x-ticks 16;num-y-ticks 5;labeloverrides 3;x-label "time (s)";y-label "position (m)";numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$3+\dfrac{1}{4}\left( x-4\right) ^{2}$};linecolor "black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "0,15";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";valid_file "T";tempfilename 'NBLTPJ01.wmf';tempfile-properties "XPR";}% } \end{enumerate} \bigskip \bigskip \bigskip \bigskip \bigskip \bigskip \bigskip \vspace{1in} \vspace{2in} \bigskip {\small %TCIMACRO{\TeXButton{\small}{\small}}% %BeginExpansion \small% %EndExpansion } \href{https://teaching.martahidegkuti.com/shared/lnotes/lecturenotes.html}{% For more documents like this, visit our page at\ https://teaching.martahidegkuti.com and click on Lecture Notes. \ E-mail questions or comments to mhidegkuti@ccc.edu.} \end{document}