Projectile Motion Lab (2D)

Without drag the motion splits into constant horizontal velocity and uniform vertical acceleration -g. Range from height h0 is R = (v cos a / g)(v sin a + sqrt(v^2 sin^2 a + 2 g h0)); from the ground it peaks at 45 degrees.

With quadratic drag the acceleration is a = -(k/m)|v| v - g y_hat, where k = 0.5 rho Cd A (same constants as the 3D lab: rho = 1.2 kg/m^3, effective area 0.016 m^2 including its tuning factor). There is no closed form, so the path is integrated with RK4 (dt = 5 ms). The best angle drops below 45 degrees and the descent is steeper than the climb. A body falling straight down approaches the terminal speed v_t = sqrt(m g / k), with v(t) = v_t tanh(g t / v_t).

Simplifications: constant Cd and air density, no wind, no spin (Magnus) force, flat ground, a point mass. The drag strength is a teaching scale, not a calibrated object.