We will use Mathematica to produce a table that shows how changing the barrel angle changes the range. Then we will study the table for conclusions about maximizing range. Basically this table is created by determining the time at which the projectile hits the ground (found in the command Solve[ y[t,theta]==0,t]) for given theta and then ascertains the x position (range) for that time.

Input := 


vnaught= 300;

x[t_, theta_]=vnaught * Cos[theta] *t; 

y[t_, theta_]=-16t^2 + vnaught* Sin[theta] *t;

Input := 


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

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

	{theta,0,Pi/2,Pi/24} ] ,

	TableHeadings->{None,{"theta","landing pt"}}]

Output =


   theta   landing pt

   0       0


   Pi

   --

   24      727.929


   Pi

   --

   12      1406.25


   Pi

   --

   8       1988.74


   Pi

   --

   6       2435.7


   5 Pi

   ----

    24     2716.67


   Pi

   --

   4       2812.5


   7 Pi

   ----

    24     2716.67


   Pi

   --

   3       2435.7


   3 Pi

   ----

    8      1988.74


   5 Pi

   ----

    12     1406.25


   11 Pi

   -----

    24     727.929


   Pi

   --

   2       0