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The science
Motion under gravity and drag
With quadratic drag the acceleration is a = (-k |v| vx, -g - k |v| vy), where k has units 1/m. Without drag (k = 0) the path is a parabola with range R = (v cosθ / g)(v sinθ + √(v² sin²θ + 2 g h)), maximal at 45 degrees for h = 0. With drag the best angle drops below 45 degrees because the slow, high part of the path loses the most speed. The equations are integrated with fourth-order Runge-Kutta, the same method as in the 3D lab.
Energy
Per unit mass, the mechanical energy lost equals the work against drag: ΔE = ∫ k v³ dt. The check panel verifies this to better than 0.02 %.
Honest simplifications: k is constant, so the air density is uniform and there is no wind, spin (Magnus force) or Earth rotation; the ground is flat.
What to try
- Press Earth with air drag and compare the green trajectory with the dashed vacuum parabola.
- Open the range graph: the red line shows the best angle, which moves below 45 degrees as drag grows.
- Choose Cliff launch and see the best angle drop towards 30 degrees.
- Choose Moon: with no air the parabola is tall and long.