11.6: Exercises
- Page ID
- 84744
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
Before you go on, you might want to work on the following exercise.
In this exercise we’ll model the descent of a skydiver, taking into account the change in drag when the parachute opens.
- Modify the penny code from this chapter to simulate the descent of a 75 kg skydiver from an initial altitude of 4000 m. The drag constant for a skydiver without a parachute is about 0.2 kg/m. What would the velocity of the skydiver be on impact?
- After opening their parachute, the velocity of the skydiver slows to about 5 m/s. Use your simulation to find the drag constant that yields a terminal velocity of 5 m/s.
- Increase the mass of the skydiver, and confirm that terminal velocity increases. This phenomenon is the source of the intuition that heavy objects fall faster; in air, they do!
- Now suppose the skydiver free falls until they get to an altitude of 1000 m before opening the parachute. How long would it take them to reach the ground?
- What is the lowest altitude where the skydiver can open the parachute and still land at less than 6 m/s (assuming that the parachute opens and deploys instantly)?
Here’s a question from the website Ask an Astronomer (see https://greenteapress.com/matlab/astro):
If the Earth suddenly stopped orbiting the Sun, I know eventually it would be pulled in by the Sun’s gravity and hit it. How long would it take the Earth to hit the Sun? I imagine it would go slowly at first and then pick up speed.
Use ode45 to answer this question. Here are some suggestions about how to proceed:
- Look up the Law of Universal Gravitation and any constants you need. I suggest you work entirely in SI units: meters, kilograms, and newtons.
- When the distance between the Earth and the Sun gets small, this system behaves badly, so you should use an event function to stop when the surface of the Earth reaches the surface of the Sun.
- Express your answer in days, and plot the results as millions of kilometers versus days.