Terminal Velocity and Viscous Transients
Terminal velocity is the constant relative speed reached when the resultant external force on a body moving through a fluid becomes zero; under Stokes drag it scales with radius squared.
Why this shows up in the exam
Sedimentation and particle sizing · Rain-drop and aerosol estimates · Viscosity determination from falling spheres
Learn the idea
Terminal speed occurs when drag plus buoyancy balances weight, while the approach to it is a force-driven transient. A falling sphere initially accelerates, but increasing drag reduces the net force. At terminal speed the forces balance, so speed stays constant even though forces remain.
🧠 Memory hook: Terminal means zero acceleration, not zero forces.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- v_t = 2 r² (rho_s-rho_f)g/(9 eta) — Stokes terminal speed of a sphere in an effectively unbounded fluid
- m dv/dt = (rho_s-rho_f)Vg - 6 pi eta r v — one-dimensional transient equation with Stokes drag and constant material properties
- v(t) = v_t[1-exp(-t/tau)] — approach from rest for linear drag, with tau set by effective inertia divided by drag coefficient
How to approach it
- 1Draw weight, buoyancy, and drag separately
- 2Set net force to zero only for terminal motion
- 3Use scaling on the complete formula before substituting numbers
Common slip-ups that cost marks
- •Omitting buoyancy from terminal-force balance
- •Assuming terminal speed is reached instantly
- •Keeping v proportional to radius squared when mass or density conditions change
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A wire 1 m long and cross-sectional area 2 mm^2 extends by 1 mm under a 200 N load. Find Young modulus.
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