The theme of the next two modules is to study a ballistics system and perform an error analysis of the said system. Therefore, we start studying a traditional ballistics problem. To quantify the sensitivity to initial conditions, we will use a $$Monte Carlo$$ approach to perform statistical analysis. Using the Gaussian normal distribution $($Bell Curve$),$ we will examine and make use of multivariate distributions. At the end of this discussion, we should be able to define the sensitivity of initial conditions on the outcome $($hitting a target$).$

Ballistics & plotting of a trajectory:

Describe one $$Ballistics device$$ and present applicable performance specifications. These might be current or historical devices, you are encouraged to be creative and add some pictures or videos if available.

MATLAB: A projectile is shot at a certain angle with respect to the horizontal at a given velocity $(100$m$/$s $<$ V $<300$m$/$s$).$ Earth's gravity is $9\mathrm{.}81$ m$/$s$^2.$ Take the variable x as the range and y as the altitude. Create a MATLAB plot, which plots that trajectory. Now, assume the terrain slopes up by $1$ m in y for every $10$ m in x$.$ On the same axis, plot the line of the terrain in a different color.

MATLAB: Continuing with the previous bullet point, use a "for" loop or a "while" loop to propagate the projectile in $0\mathrm{.}01$ second time steps all the way until impact. Mark the impact spot with a marker and display the impact coordinates on the projectile$/$terrain plot. Label the axes accordingly and give your figure a title. While keeping the initial velocity constant, "experimentally" determine the initial angle of the projectile with respect to the horizontal in order to achieve maximum range. Is there an analytical solution? Share that result with your fellow students.