Homework #7

Homework #7#

Completing this assignment

Upload your C++ source file (not the executables produced by g++) and the plot output (a PNG file) to Brightspace.

Important

Make sure your that g++ can compile your code. You may need to use -std=c++20.

Important

All work must be your own.

You may not use generative AI / large-language models for any part of this assignment.

  1. Projectiles with air-resistance :

    Let’s try solving for projectile motion with drag (air resistance).

    We want to solve:

    \[\begin{split}\dot{\bf r} &= {\bf v} \\ \dot{\bf v} &= \frac{1}{m} {\bf F}_a({\bf v}) - |g| \hat{\bf y}\end{split}\]

    where \({\bf r} = (x, y)\) is the position vector, \({\bf v} = (v_x, v_y)\) is the velocity vector, \(m\) is the mass of the projectile and

    \[{\bf F}_a({\bf v}) = - \frac{1}{2}C \rho_\mathrm{air} A |v| {\bf v}\]

    is the Newton drag term that is applicable for large Reynolds numbers (turbulent), and \(\rho_\mathrm{air}\) is the density of air, \(C\) is the drag coefficient, \(A\) is the cross-sectional area of the projectile.

    Note

    Gravity only affects the vertical (\(y\)) component of velocity, but drag affects both, since it is proportional to \({\bf v}\).

    We’ll consider a baseball. Then we can take:

    • \(C = 0.3\)

    • \(A = \pi (d/2)^2\) with the diameter of a bassball, \(d = 0.074~\mathrm{m}\)

    • \(m = 0.145~\mathrm{kg}\)

    • \(\rho_\mathrm{air} = 1.2~\mathrm{kg~m^{-3}}\)

    and we take the gravitational acceleration to be \(g = -9.81~\mathrm{m~s^{-2}}\),

    We’ll imagine throwing the baseball from some height, \(y_0\), above the ground (\(y = 0\)) at an angle \(\theta\) from the horizontal with velocity magnitude \(v_0\).

    This means our initial conditions are:

    \[\begin{split}x_0 &= 0 \\ y_0 &= y_0 \\ {v_x}_0 &= v_0 \cos(\theta) \\ {v_y}_0 &= v_0 \sin(\theta)\end{split}\]

    and we want to integrate until the ball hits the ground, \(y(t_\mathrm{max}) = 0\).

    Your task

    Solve this system using our 2nd-order Runge-Kutta method.

    You should structure your program the same way we did for our 2nd-order Runge-Kutta Orbit code in class. The main differences will be to the righthand side function and the stopping criterion.

    Run the case with and without drag (you can set \(C = 0\) for no drag), and plot the results together on the same plot.

    Tip

    If you have 2 output files, file1.txt and file2.txt, you can plot both of them together in gnuplot as:

    plot 'file1.txt' using 1:2 title "no drag", 'file2.txt' using 1:2 title "drag"

    This will plot column 2 vs. column 1 from each file as separate lines and label them in the plot.