Projectile Motion – Numerical Solution of the Equations of Motion

Equations of motion. Newton's second law gives a set of four first-order ODEs for s = (x, y, vₓ, v_y): dx/dt = vₓ, dy/dt = v_y, dvₓ/dt = aₓ, dv_y/dt = a_y − g. Drag acceleration: a = 0 (A), a = −γ·v (B), a = −c·|v|·v (C). The coefficient c can be interpreted as c = ρ·C_D·A/(2m).

Euler method: s_n+1 = s_n + Δt·f(s_n, t_n). It is simple but only first-order accurate: the global error scales as Δt.

RK4: evaluates four slopes k₁…k₄ and combines them: s_n+1 = s_n + (Δt/6)(k₁ + 2k₂ + 2k₃ + k₄). The global error scales as Δt⁴, so for a sufficiently small step it is far more accurate.

Impact is not taken as the first point below ground: between the last point above ground and the first point below it, the position is interpolated with a cubic Hermite polynomial and the root y = 0 is found by bisection.

Analytical vs. numerical solution. Without drag and with linear drag an exact solution exists (shown dashed; for γ → 0 the no-drag limit is used). For quadratic drag the equations are coupled and are solved numerically – no elementary closed-form trajectory is used here.

1. No air resistance, Euler, Δt = 0.5 s: compare the numerical and analytical trajectories.
2. Reduce Δt step by step: 0.5, 0.2, 0.1, 0.05, 0.01 s, and watch how the error changes.
3. Repeat with RK4.
4. Compare the motion on Earth, the Moon and Mars (comparison mode).
5. Compare trajectories with and without air resistance.
6. Without drag: do complementary angles such as 30° and 60° give equal ranges?
7. Repeat the previous experiment with air resistance.
8. How does the importance of quadratic drag change as the initial speed increases?