Back

Free fall under vacuum, Stokes drag, and quadratic drag

What you are seeing: three identical balls dropped from the same height at t=0t = 0. The yellow one experiences no drag (vacuum, v˙=g\dot v = -g); the cyan one experiences Stokes drag (v˙=gbv\dot v = -g - b v, linear in velocity); the orange one experiences quadratic drag (v˙=gcvv\dot v = -g - c |v| v). They land at different times and arrive with different velocities.

The right-side plot tracks v(t)|v(t)| for each ball. Vacuum is a straight line v=gt|v| = g t. Stokes asymptotes to its terminal vt=mg/bv_t = mg/b exponentially. Quadratic asymptotes to vt=mg/cv_t = \sqrt{mg/c} on a slower-than-exponential approach (think tanh\tanh shape). Crossover between the two drag laws happens at vc=b/cv_c = b/c: above this speed quadratic drag dominates, below it Stokes dominates.

Figure 1. Free fall under three drag laws. Method: RK4 at dt = 1/240.
y_0 (m)120
b (Stokes)0.50
c (quad)0.050

WHAT TO TRY

  • Vary each control and watch the rail readouts respond.
  • Compare the diagnostic plot against the live scene.