why , regardless of the magnitude of the initial velocity, the maximum horizontal displacement for a projectile occurs when the angle is 45 degrees. Specifically, what happens when the angle is increased or decreased. How would you mention vector components and also trigonometric concepts into the response?
If you solve the two equations (x and y)of projectile motion to find range [in y direction t = 2 vo sin(theta)/g which substitutes into the x direction R = vo cos (theta) t, then use trig identity 2 sin(theta)cos(theta) = sin(2 theta)] you get
R = v sin(2 theta)/g.
At 45 degrees sin (2 theta) = 1 which is the maximum value it can have.