In the analysis that follows, we derive the equation of motion of a projectile as measured from an inertial frame at rest with respect to the ground. Associated with the frame is a right-hand coordinate system with its origin at the point of launch of the projectile. The x -axis is tangent to the ground, and the y axis is perpendicular to it (parallel to the gravitational field lines). Let g be the acceleration of gravity. Relative to the flat terrain, let the initial horizontal speed be v h = v cos ⁡ ⁡ ( θ θ ) and the initial vertical speed be v v = v sin ⁡ ⁡ ( θ θ ) . It will also be shown that the range is 2 v h v v / g , and the maximum altitude is v v 2 / 2 g . The maximum range for a given initial speed v is obtained when v h = v v , i.e. the initial angle is 45 ∘ ∘ . This range is v 2 / g , and the maximum altitude at the maximum range is v 2 / ( 4 g ) .