A particle starts from rest and obeys dv/dt = α-βv, where α,β>0. Find the displacement by the time its speed becomes half of its terminal speed.
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A particle starts from rest and obeys dv/dt = α−βv, where α,β>0. Find the displacement by the time its speed becomes half of its terminal speed.

G1 Physics · Accepted Answer

Terminal speed is vₜ = α/β. Use a = v dv/dx = α−βv. Thus dx = v dv/(α−βv). Let y=βv/α. Then v=(α/β)y and dv=(α/β)dy. Hence dx = (α/β²) y/(1−y) dy. The required change is y: 0 → 1/2. Therefore x = (α/β²)∫₀→1/2 y/(1−y) dy = (α/β²)[−y−ln(1−y)]₀→1/2 = α(ln2−1/2)/β².