PHYS 105:  Introduction to Computational Physics

Spring 2008

Homework #4
(Due:  May 15, 2008)



  1. Reconsider the 1-dimensional motion of a particle moving under a harmonic force:

    \begin{displaymath} a(x) = -k\,x\,, \end{displaymath}

    again with $k = 4$ and $x = 0$, $v = 1$ at $t = 0$. However, now friction also acts on the particle, producing an acceleration proportional to, and always opposite, the velocity:

    \begin{displaymath} a_f = -\alpha v\,. \end{displaymath}

    For small values of $\alpha$, the mathematical solution to the equations of motion may be shown to be of the form

    \begin{displaymath} x = A\,e^{-bt}\,\sin\omega^\prime t\,. \end{displaymath}

    (a) For $\alpha = 0.1$, and taking time steps $\delta t = 0.001$, determine the period $T$ and hence $\omega^\prime$ by finding the first time (after $t=0$) the particle crosses $x=0$ with $v > 0$ (using linear interpolation, as usual, to refine the answer).

    (b) By considering the decrease in amplitude from one maximum (or minimum) to the next, determine the ratio $r$ by which the amplitude decreases from one peak to the next and hence the value of $b$ for this $\alpha$.


  2. Do in-class exercise 6.2. Turn in your program (parts 1 and 2), three plots (parts 3-5), and clearly state the value of the slope you obtain in part 5.


  3. (a) A particle moves in two dimensions under the combined effects of gravity and air resistance. The components of its acceleration are


    $\displaystyle a_x$ $\textstyle =$ $\displaystyle -\alpha\,v_x\,,$  
    $\displaystyle a_y$ $\textstyle =$ $\displaystyle -g -\alpha\,v_y\,,$  

    where $g = 9.8 {\rm m/s}^2$. The particle is launched at time $t=0$ from $x=0, y=0$ with a speed of $v_0 = 100$ m/s at an angle $\theta$ to the horizontal.

    It is desired to have the projectile hit a horizontal target running from $x=300$ to $x=320$ m at a height of $y=200$ m, from above -- i.e. with $v_y < 0$. (Note: To determine whether this occurs, first interpolate the trajectory to the value of $x$ when $y=200$ m, then check whether $x$ lies in the desired range.)

    (a) In the case of no air resistance ($\alpha = 0$), determine, to one decimal place, the minimum and maximum values of $\theta$ in degrees that will accomplish the goal. You may do this numerically or analytically. If you do it numerically, find the angle to within $0.1^\circ$.

    (b) For $\alpha = 0.1$, can the goal still be accomplished for the given value of $v_0$? If the answer is yes, then determine the range of $\theta$ (to an accuracy of $0.1^\circ$) that will hit the target, as in part (a). If no, determine instead the minimum value of $v_0$ needed to reach the target. Use a time step of $\delta t$ = 0.1.

    (c) Repeat part (b) for $\alpha = 0.25$.


As usual, in all cases, turn in your program, the output produced when it runs, and any requested plots and additional calculations.






Steve McMillan 2008-05-08