A particle is projected vertically upward with speed u in a medium where the resistance per unit mass during ascent is βv. If βu/g ≪ 1, the maximum height up to first order in β is approximately:
Options

Question

A particle is projected vertically upward with speed u in a medium where the resistance per unit mass during ascent is βv. If βu/g ≪ 1, the maximum height up to first order in β is approximately:

G1 Physics · Accepted Answer

From the exact result, H = u/β − (g/β²)ln(1+βu/g). Let z = βu/g ≪ 1. Use ln(1+z) = z − z²/2 + z³/3 + ... Then H = u/β − (g/β²)[z − z²/2 + z³/3 + ...] The first term cancels the z-term, giving H ≈ u²/(2g) − βu³/(3g²). Therefore option A is correct.