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Science / Forces / Undergraduate / sfd40

Falling Speeds: Solve the Linear-Drag Model

Derive a time-dependent fall from a first-order differential equation.

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5 questions0m 0s

Prerequisites

First-order differential equations, Newton second law, units and exponential functions.

Learn the skill

Take downward velocity positive, constant g, fixed mass m and linear drag b v in still fluid. Ignore buoyancy. Then m dv/dt = mg - b v, tau = m/b and vT = mg/b. From rest, v(t) = vT(1 - exp(-t/tau)). Position y(t) = vT[t - tau(1 - exp(-t/tau))] with y(0) = 0.

Worked example

For m = 2 kg, b = 1 kg/s and g = 10 m/s^2, tau = 2 s and vT = 20 m/s. At t = tau, v/vT = 1 - exp(-1), about 0.632. Use linear drag only in a suitable regime.

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 What is tau for the example?
Question 2 What is the model vT?
Question 3 At t = tau, v/vT is approximately what?
Question 4 At t = tau, what is y to two decimal places?
Question 5 If b doubles with m and g fixed, what happens?

Further inquiry

Derive velocity with an integrating factor and integrate once more for displacement. Check the rest-start and long-time limits, then solve for a nonzero initial downward velocity. Explain when the linear-drag assumption and ignored buoyancy are unsuitable.

Review criteria

  • State the sign convention and both initial conditions.
  • Check dimensions of b, tau, speed and displacement.
  • Separate a mathematical drag model from the validity of a physical regime.