A particle is projected vertically upward with speed u in a medium where the air resistance per unit mass is γv² and is always opposite to velocity. Define terminal speed V=√(g/γ). The time of ascent and maximum height are:
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Question

A particle is projected vertically upward with speed u in a medium where the air resistance per unit mass is γv² and is always opposite to velocity. Define terminal speed V=√(g/γ). The time of ascent and maximum height are:

Options

G1 Physics · Accepted Answer

During ascent, dv/dt = −g − γv². Since V² = g/γ, dv/dt = −g[1+v²/V²]. Thus ascent time = ∫₀→u dv / {g[1+v²/V²]} = (V/g)tan⁻¹(u/V). For the height, use a = v dv/dx: v dv/dx = −g[1+v²/V²]. Hence H = ∫₀→u v dv / {g[1+v²/V²]} = (V²/2g)ln[1+(u/V)²]. Therefore option A is correct.