because we did not work with C or Fortran language and compiler yet. But if
you're learning C or F, we'll be very impressed to receive your calculations
(in Quercus Assignments) done in one of those languages!
Write a procedure that takes $\alpha$ as input, and returns:
(i) maximum height $h$ in meters,
(ii) the range $R$ in meters [the ball touches down at point $(R,0)$] and
(iii) the ratio of $x_h/R$ (to judge the asymmetry of trajectory's shape).
Here, $x_h$ is the $x$ value at which the height is maximum.
If the motion takes place in vacuum, the problem has a simple analytical solution for (i)-(iii). Please derive it. Show that, in vacuum, maximum range $R$ is obtained with $\alpha_m = 45^\circ$ (evaluate $R$ and $h$ in vacuum), and the trajectory is a symmetric, inverted parabola ($x_h/R = 1/2$).
Numerical calculation is necessary when the tennis ball moves in air of standard atmospheric density $\rho = 1.225$ kg/m$^3$. Do not try to find analytical solution, since it is not known, except for very special $\alpha$, like 0 or 90$^\circ$. Please integrate the equations of motion to simulate the motion until the touchdown point, and explain your choice of method any other details you find important. It should be simple but it has to be accurate, so we suggest timestep dt=0.001 s. You are asked to code it yourself (do not use black-box packages or modules to do the simulation). With $\vec{r} = (x,y)$ being the position vector of the ball and $\vec{v}=\dot{\vec{r}}$ its velocity vector, the equation of motion reads $$ \ddot{\vec{r}} = -g\, \hat{y} -\frac{C_d A}{2M} \rho\, v \; \vec{v}$$ where: $v = |\vec{v}|$, $\hat{y} $ is a vertical versor pointing upward, $C_d=0.55$ is a nondimensional drag coefficient of tennis balls, $A$ is the cross sectional area of the ball, $M$ its mass, and $g = 9.81$ m/s$^2$.
You will code the problem as either two coupled 2nd order ODEs in variables $x$ and $y$ or four coupled 1st order ODEs for position and velocity components. The above vector eq. of motion splits into two separate 2nd order ODEs in two directions, while the 1st order eqs. are obtained by writing first writing $\ddot{\vec{r}} = \dot{\vec{v}}= ...$, plus another vector equation $\dot{\vec{r}} = \vec{v}$, and then splitting each into components). Test your procedure on a problem with $\rho=0$, for which you know the answer, and see to it that the air drag changes that solution, but not qualitatively (e.g., make sure the tennis ball doesn't travel a few meters only, or shoots to the Moon with increasing speed, all of which would indicate a programming bug or worse yet, a correct solution of wrong equations).
With the tested procedure, find the angle $\alpha_m$ which results in maximum range $R$. What are the quantities (i)-(iii) then? Plots of trajectories are welcome!
You are given a series of values $x$ of length $N$; that is a vector of values $x_i$, where $i=1...N$. Formulate an algorithm that finds the maximum drawdown (maximum drop) of $x$ (representing investment or voltage) in some contiguous segment of data, i.e. the largest drop of value from some initial value $x_i$ to a lower value $x_j$ at index $j > i$.
For instance, if the series is $x$ = {4, 3, 6, 5, 4, 5, 2, 3, 4, 4, 11, 8, 10} then the largest dropdown is between positions $i=3$ and $j=7$, and equals $x_3-x_7= 6-2=4$. Notice that we could have even more points between those two, with values oscillating between 2 and 6, and the resuting maxmium dropdown would not change. A different $j$ might result but we are not interested in $i$ or $j$, just the maximum drop (precise interval can be provided with very little extra work, but the problem does not ask for it). Your algorithm must thus somehow not be fooled by local extrema.
You can describe your algorithm in words, and/or implement it, commenting the program very well, so it is clear how it works and how you proved that it correctly works to yourself.
Remark: Since all contiguous subsets (subintervals) of the dataset provide possible solution, examining them all separately would take an effort of computational complexity $O(N^2)$, because the number of all index pairs {$j\gt i$} is $N(N-1)/2$, so $\sim N^2$. For large $N$ this is an awful lot of computation. Try to obtain a cheaper asymptotic scaling with $N$ if you can. If you can't, code an inefficient method scaling like $N^2$, for a partial credit of up to 13 pts.
Show it numerically for a particular choice of function: $$ y(x) = e^{x-1} + x^3(x - 1) $$ around the point $x=1$. Make a plot of the difference between true analytical second derivative and its estimate $f$ above, as a function of $h$.
The plot should have a log-log scaling to reveal any power-law dependencies between the error and $h$. (What is the slope?) The suggested range of $h$ should be from, say, $10^{-7}$, to $10^{-2}$. and the power of ten that produces $h$ (not $h$ itself) should cover that range uniformly. Otherwise there will be a pileup of points toward one end of the plot in log-log axes.