Independent horizontal and vertical motion
With no air resistance the horizontal velocity stays u cos θ. The vertical velocity starts at u sin θ and changes by −gt. The simulation uses x = (u cos θ)t and y = (u sin θ)t − ½gt².
Set a launch speed and angle and the page draws the whole flight onto level ground. With drag at zero the path is the exact vacuum trajectory. Advanced mode adds quadratic air resistance and integrates the motion step by step. Gravity can be changed from a Moon-like 1.6 m/s² up to a much stronger field. The projectile is a point mass.
With no air resistance the horizontal velocity stays u cos θ. The vertical velocity starts at u sin θ and changes by −gt. The simulation uses x = (u cos θ)t and y = (u sin θ)t − ½gt².
On level ground with no drag, the time of flight is 2u sin θ / g, the range is u² sin 2θ / g, and the greatest height is (u sin θ)² / (2g). Range is greatest at 45° because sin 2θ is greatest there.
sin 2θ is the same for 30° and 60°, so the ranges match when drag is zero. The steeper launch spends more of its speed vertically, so it goes higher and takes longer to return.
Quadratic drag opposes the velocity and is larger at higher speed. The projectile therefore slows horizontally as well as vertically, the descending path is steeper than the climb, and range is shorter. The best angle is then less than 45° because a flatter launch keeps the speed, and the drag, a little lower.
GCSE treats projectile motion as constant horizontal velocity plus vertical acceleration g, often with 45° as the angle for maximum range. A level and IB add the resolved equations and, as an extension, the idea that drag removes the simple 45° result.
The ground is level, gravity is constant, and the vacuum path ignores the size of the object. The drag option assumes still air and a force proportional to speed squared. It does not include spin, lift or a changing air density.