A body is dropped from rest in a medium where resistance per unit mass is βv. If its terminal speed is V=g/β, the distance fallen by the time its speed becomes V/2 is:
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A body is dropped from rest in a medium where resistance per unit mass is βv. If its terminal speed is V=g/β, the distance fallen by the time its speed becomes V/2 is:

G1 Physics · Accepted Answer

Take downward as positive. The equation is dv/dt = g−βv. With v(0)=0, v = V(1−exp(−βt)), where V=g/β. When v=V/2, exp(−βt) = 1/2 ⇒ t = ln2/β. Distance fallen: x = ∫₀→t V(1−exp(−βt))dt = V[t + (exp(−βt)−1)/β]. Substituting t=ln2/β and exp(−βt)=1/2, x = (g/β²)(ln2−1/2).