PHYS 105:  Introduction to Computational Physics

Spring 2008

Homework #5
(Due:  May 22, 2008)



  1. Do in-class exercise 7.1. Turn in your program (part 2) and three plots (parts 4 and 5), and clearly state the results you obtain in response to the other questions asked.

  2. Consider again a projectile moving in two dimensions under the combined effects of gravity and air resistance. Initially the projectile is launched from $x=0, y=0$ with speed $v_0 = 100$ m/s at an angle of $\theta_0=60^\circ$ to the horizontal. The components of its acceleration are


    $\displaystyle a_x$ $\textstyle =$ $\displaystyle -\beta\,\vert v\vert\,v_x\,,$  
    $\displaystyle a_y$ $\textstyle =$ $\displaystyle -g -\beta\,\vert v\vert\,v_y\,,$  

    where $\beta = 0.001$ (chosen so that the initial acceleration for this value of $v_0$ is the same as the case $\alpha=0.1$ in problem 1).

    (a)
    Compute the projectile's range and time of flight (take $\delta t = 0.01$, and don't forget to interpolate!), and compare them to the results of the $\alpha=0.1$ calculation in problem 1. Plot both trajectories ($\alpha=0.1$ and $\beta=0.001$) on the same graph.

    (b)
    At what angle to the horizontal $\theta_1$ does the projectile hit the ground in each case? What would $\theta_1$ be in the absence of air resistance (i.e. $\alpha=\beta=0$)?

    (c)
    By what factor (to within 1 percent) must the launch speed be increased to restore the range to the $\beta=0$ result?

    (d)
    By varying the value of $\theta_0$, determine the maximum range of the projectile for $v_0 = 100$ m/s. To what value of $\theta_0$ (to 1 decimal place) does this correspond?

    (e)
    For $v_0 = 100$ m/s, plot $\theta_1$ as a function of $\theta_0$.

  3. Now suppose that the value of $\beta$ in problem 2 varies with height $y$, according to the law


    $\displaystyle \beta(y)$ $\textstyle =$ $\displaystyle 0.001 e^{-y/h}\,,$  

    where $h$ = 500 m (not a very realistic description of Earth's atmosphere!). How does the maximum range of the projectile (as computed in problem 2d) change as a result? What if $h$ = 5 km (a much better approximation to reality)?






Steve McMillan 2008-05-12