The date is 1750 and Baron Munchausen, the scourge of Thuringia, has accused you of being a spy for the nearby state of Hesse. He has expressed a willingness to have you shot without trial, but has also said that he might change his mind if you can solve a problem that has baffled his gunnery officers. Based on your calculus expertise (and the fact that your life is on the line) you agree to try.
The Baron's problem appears to be relatively simple. It is well known that an angle of 
provides the greatest projectile range when the ground is flat. However, this angle does not seem to work as well if the Baron's artillery batteries are forced to fire uphill or downhill. The Baron wants to know what the optimum angle is in these cases.
You, with your great knowledge of projectile trajectories and calculus, immediately begin to set the problem up. You first realize that setting up your coordinates as in Figure 
will make it easier, because 
will be the distance that the Baron is interested in at the value of t where 
. In setting up your coordinates this way, you introduce two angles, 
and 
where 
is the angle between the cannon and the ground and 
is the angle the ground makes with the horizontal.
Given this coordinate system, you derive the following set of differential equations
and initial conditions
where 
is the initial velocity of the projectile.
Your assignment is to find the angle 
that gives the maximum range for each value of 
. In writing your report, the following points must be addressed.
. Note that the variables 
, 
, and 
will appear as parameters in your solution.
that gives maximum range for a given angle 
. A plot of 
versus 
should be provided for the use of the Baron's field officers.
Schedule for Project 2
