A sphere is released from rest in a viscous liquid and obeys Stokes drag. Its terminal speed is vₜ and its Stokes time constant is τ. What distance does it travel before its speed reaches vₜ/2?
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Question

A sphere is released from rest in a viscous liquid and obeys Stokes drag. Its terminal speed is vₜ and its Stokes time constant is τ. What distance does it travel before its speed reaches vₜ/2?

G1 Physics · Accepted Answer

The speed is v(t) = vₜ[1−exp(−t/τ)]. Integrate with respect to time from 0 to t: x(t) = vₜ[t−τ(1−exp(−t/τ))]. The speed becomes vₜ/2 at t = τ ln2. At this instant, exp(−t/τ) = 1/2. Therefore, x = vₜ[ τln2 −τ(1−1/2) ]. Thus, x = vₜτ(ln2−1/2). Correct option: B.