Exercise 1.1.30

Question

An airplane takes off from an airport and lands an hour later at another airport, 400 miles away. If t represents the time in minutes since the plane has left the terminal building, let $x(t)$ be the horizontal distance traveled and $y(t)$ be the altitude of the plane.
    \begin{enumerate}[label=(\alph*)]
        \item Sketch a possible graph of $x(t)$.
        \item Sketch a possible graph of $y(t)$.
        \item Sketch a possible graph of the ground speed.
        \item Sketch a possible graph of the vertical velocity.
    \end{enumerate}

Khagan Gulusoy · Accepted Answer

\documentclass[10pt]{article}
\usepackage[T1]{fontenc}
\usepackage[english]{babel}
\usepackage{enumitem}
\usepackage{float}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}

\begin{document}
For simplicity, we will neglect the portion of the journey that involves the airplane traveling on the ground.
\begin{enumerate}[label=(\alph*)]
\item When the airplane takes off, it obviously ascends. Since it doesn't only fly vertically (otherwise it would be a rocket), the horizontal distance travelled by the plane gradually increases along with the vertical distance. Afterwards, the plane starts flying horizontally, causing a faster rise in the horizontal distance travelled. When the airplane descends for landing, the horizontal distance gradually increases for the same reason as during takeoff. We can sketch the graph of the function $x$ to approximately resemble the following:
  \begin{figure}[H] \centering
    \begin{tikzpicture}[scale=0.90]
      \begin{axis}[axis lines=center, width=10cm, height=4cm, xmin=0, xmax=3.3, ymin=0, ymax=2.5, xtick={3}, ytick=\empty, xticklabel={60 min}, xlabel={$t$, time}, ylabel={$x(t)$, horizontal distance}, x label style = {at=(ticklabel cs:0.5),anchor=north}, y label style={at=(ticklabel cs:0.5),rotate=90,anchor=south}]
        \addplot[domain=0:3, samples=150, mark=none, color=red, ultra thick] {sin(deg(x-1.5)) +1};
      \end{axis}
    \end{tikzpicture}
  \end{figure}
  
\item 	extit{For the sake of simplicity, we assume that the two cities are at the same level above sea level.}
  When the airplane first starts its takeoff, the altitude of the plane increases very slowly. Afterward, the altitude increases considerably. Once the airplane stops ascending and begins moving horizontally, the height remains unchanged. As the plane descends, the height obviously decreases. Upon landing, the altitude gradually decreases. We can roughly sketch the graph of $y$ as follows:

\begin{figure}[H] \centering
    \begin{tikzpicture}[scale=0.90]
      \begin{axis}[width=10cm, height=4cm, axis lines=center, xmin=0, xmax=5.5, ymin=0, ymax=1.5, xtick={5.277}, ytick=\empty, xticklabel={60 min}, xlabel={$t$, time}, x label style = {at={(ticklabel cs:0.5)},anchor=north}, ylabel={$y(t)$, altitude}, y label style={at=(ticklabel cs:0.5),rotate=90,anchor=south}]
        \addplot[domain=0:1.292, samples=50, mark=none, color=red, ultra thick] {sin(deg(pi/2*x))*x}; 
        \addplot[domain=1.292:4, samples=2, mark=none, color=red, ultra thick] {1.158}; 
        \addplot[domain=4:5.277, samples=100, mark=none, color=red, ultra thick] {1.158 - sin(deg((pi/2)*(x - 4)))*(x - 4)}; 
      \end{axis}
    \end{tikzpicture}
  \end{figure}
  
\item After the plane flies off, its ground speed increases before it reaches some particular height and starts moving horizontally. After the airplane starts descending for landing, the ground speed will considerably decrease in the vicinity of the airport.

\begin{figure}[H] \centering
    \begin{tikzpicture}[scale=0.90]
      \begin{axis}[width=10cm, height=4cm, axis lines=center, xmin=0, xmax=19, ymin=0, ymax=14, xlabel={$t$, time}, ylabel={$v_g$, ground speed}, y label style={at=(ticklabel cs:0.5),rotate=90,anchor=south}, xtick={17.642}, ytick=\empty, xticklabel={60 min}, x label style = {at={(ticklabel cs:0.5)},anchor=north}]
        \addplot[domain=0:2, samples=50, mark=none, color=red, ultra thick] {0.5*sin(deg(x-1.5)) +0.499}; 
        \addplot[domain=2:5.15, samples=150, mark=none, color=red, ultra thick] {6*sin(deg(x-pi/2-2))+6.739};
        \addplot[domain=5.15:13, samples=2, mark=none, color=red, ultra thick] {12.739};
        \addplot[domain=13:16.142, samples=150, mark=none, color=red, ultra thick] {6*sin(deg(x+4.5*pi-13))+6.739};
        \addplot[domain=16.142:17.642, samples=50, mark=none, color=red, ultra thick] {1/1.35*sin(deg(x+5*pi-16.14))+0.739};
      \end{axis}
    \end{tikzpicture}
  \end{figure}
  
\item After the airplane takes off, its vertical speed starts increasing. In order to initiate horizontal motion, the plane reduces its vertical speed, which remains constant while flying horizontally. As the plane begins descending, the vertical speed decreases. During landing, the pilot increases the plane's vertical speed from negative to zero.

\begin{figure}[H] \centering
    \begin{tikzpicture}[scale=0.90]
      \begin{axis}[axis lines=center, width=10cm, height=6cm, xmin=0, xmax=22, ymin=-4.2, ymax=4.2, ytick=\empty, y label style={anchor=east}, xtick={21.495}, xticklabel={60 min}, xlabel={$t$, time}, ylabel={$v$, vertical speed}, x label style = {at={(ticklabel cs:0.5)},anchor=north}, y label style={at=(ticklabel cs:0.5),rotate=90,anchor=south}]
        \addplot[domain=0.929:7.212, samples=150, mark=none, color=red, ultra thick] {2*sin(deg(x-2.5))+2};
        \addplot[domain=7.212:15.21, samples=150, mark=none, color=red, ultra thick] {0};
        \addplot[domain=15.21:21.495, samples=150, mark=none, color=red, ultra thick] {2*sin(deg((x-(2.5*pi)-12.07)))-2};
      \end{axis}
    \end{tikzpicture}
  \end{figure}
\end{enumerate}
\end{document}

Exercise 1.1.30

Answers

Comments

Exercise 1.1.30

Answers

Comments

Add answer