Table approach

The first table produced in Part 2 showed that we can not only hit the target, we can overshoot it. However, there is a limit to how much we can increase our range by moving back the hill - say with an angle of elevation of Pi/4 radians or 45 degrees. The target will be unreachable beyond this maximum range. Expanding our table from Part 2, we get

Input := 


theta = Pi/4;

Input := 


TableForm[Table[ {k, x[t,k]/.

	Solve[ y[t,k]==0,t] [[2]] //N }, 

	{k,0,5000,200} ] ,

	TableHeadings->{None,{"x-coord","landing pt"}}]

Output =


   x-coord   landing pt

   0         2812.5


   200       2924.34


   400       2987.69


   600       3019.93


   800       3030.02


   1000      3023.3


   1200      3003.25


   1400      2972.31


   1600      2932.23


   1800      2884.37


   2000      2829.75


   2200      2769.2


   2400      2703.4


   2600      2632.89


   2800      2558.14


   3000      2479.54


   3200      2397.43


   3400      2312.09


   3600      2223.78


   3800      2132.71


   4000      2039.08


   4200      1943.06


   4400      1844.81


   4600      1744.46


   4800      1642.14


   5000      1537.97

This tells we can make a projectile land at (3030, 0) by moving our catapult back and up to (-800, 800 Sqrt[3]). This is close to our maximum range. Even if we move the catapult higher, eventually the effect of gravity will accelerate it toward earth before it has a chance to increase its range.

We continue to isolate the optimal placement by focusing in on the interval from 750 to 850.

Input := 


TableForm[Table[ {k, x[t,k]/.

	Solve[ y[t,k]==0,t] [[2]] //N }, 

	{k,750,850,10} ] ,

	TableHeadings->{None,{"x-coord","landing pt"}}]

Output =


   x-coord   landing pt

   750       3029.24


   760       3029.48


   770       3029.68


   780       3029.84


   790       3029.95


   800       3030.02


   810       3030.05


   820       3030.03


   830       3029.98


   840       3029.89


   850       3029.75

From this table (3030.05, 0) would appear to be the farthest point we could hit on the plane where the enemy camps - assuming we still place our catapult so that its angle of elevation with the horizontal is Pi/4 radians or 45 degrees. We get this range by moving the catapult to position (-810, 810 Sqrt[3]).