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Falling Speeds: Analyze Quadratic-Drag Relaxation

Nondimensionalize a nonlinear fall and test its limiting behavior.

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5 questions0m 0s

Prerequisites

Nonlinear differential equations, nondimensionalization, hyperbolic functions and asymptotic analysis.

Learn the skill

With downward v >= 0, constant k, m, g, still fluid and no buoyancy, m dv/dt = mg - k v^2. Define vT = sqrt(mg/k), s = gt/vT and w = v/vT. From rest, dw/ds = 1 - w^2 and w = tanh(s). Displacement is y = (vT^2/g) ln(cosh(s)), measured downward from release.

Worked example

For m = 2 kg, g = 10 m/s^2 and k = 0.2 kg/m, vT = 10 m/s and s = t/(1 s). At s = 1, v/vT = tanh(1), about 0.762. Coefficients and buoyancy cannot silently change during the calculation.

Normalized speed w approaches one. For linear drag, w = 1 - exp(-s). For quadratic drag, w = tanh(s). Here s = gt/vT for each model; the dashed linear curve rises more slowly at the same normalized time.
Normalized speed w approaches one. For linear drag, w = 1 - exp(-s). For quadratic drag, w = tanh(s). Here s = gt/vT for each model; the dashed linear curve rises more slowly at the same normalized time.
Question 1 At s = 1, what is w to three decimals?
Question 2 When w = 0.8, what is downward acceleration as a fraction of g?
Question 3 At s = 1 for the example, what is y to two decimals?
Question 4 How much normalized time s is needed for w = 0.95, to three decimals?
Question 5 Which change invalidates this constant-coefficient closed form?

Further inquiry

Derive the dimensionless equation and its rest-start solution. Obtain displacement by integrating velocity, then recover the short-time free-fall limit. Linearize near w = 1 to find the decay rate of a small speed deficit. Discuss why varying density, drag regime or buoyancy requires revising the model.

Review criteria

  • Carry sign, coefficient and initial-condition assumptions through the derivation.
  • Show both short-time and terminal-limit checks, including the deficit decay rate.
  • Distinguish the constant-coefficient result from a realistic atmospheric descent.