Marble Trajectory Analysis Tool (2D)

Launch

Launch angle 42°

Launch speed 22.0 m/s

Marble & Air

Drag coefficient (C_d) 0.45

Marble radius 1.20 cm

Marble mass 28.0 g

Live Readout (real trajectory)

Time of flight

0.00 s

Max height

0.00 m

Range

0.00 m

Range lost to drag

0.0%

How it works

The real marble is pushed by gravity and quadratic air drag, whose force always opposes velocity and scales with speed squared:

k = 0.5 · ρ · Cd · A / m
ax = −k·|v|·vx
ay = −g − k·|v|·vy

where ρ is air density, A = πr² the marble's cross-section and m its mass. This pair of coupled equations has no closed-form solution — it's integrated here with 4th-order Runge–Kutta (RK4) at a fixed 2 ms sub-step. Because drag removes more energy on the way up (where speed is highest right after launch) than gravity alone would, the real trajectory is asymmetric: the descent is steeper and shorter than the climb, unlike an ideal parabola.

The dashed curve is the textbook drag-free parabola computed from the closed-form formulas for comparison:

h_ideal = v0²sin²θ / (2g)
R_ideal = v0²sin(2θ) / g
t_ideal = 2·v0·sinθ / g

Solid line — real trajectory (numerically integrated, gravity + drag).
Dashed line — ideal parabola (same launch angle & speed, no drag).
A heavier or larger marble changes k = drag/mass: bigger radius increases drag area, more mass dilutes it — try both to see they push the divergence in opposite directions.

Real trajectory (gravity + drag, RK4) Ideal parabola (no drag)