Verify that the launch angle \(\theta\) that produces the maximum range corresponds to \(45^{\circ}\). Verify further that in the absence of friction, although the trajectory changes, the range remains the same for angles that differ by the same amount (positive or negative) from the angle \(45^{\circ}\). Verify this numerically for a projectile fired at an angle \(\theta=30^{\circ}\) or \(\theta=60^{\circ}\). In the case of the presence of friction with the atmosphere, for which angle between \(30^{\circ}, 45^{\circ}\) and \(60^{\circ}\) would the range be higher?