A particle is projected vertically upward with speed u. During upward motion, the air resistance per unit mass is βv, always opposite to velocity, where β>0. The time of ascent and maximum height are respectively:
Options

Question

A particle is projected vertically upward with speed u. During upward motion, the air resistance per unit mass is βv, always opposite to velocity, where β>0. The time of ascent and maximum height are respectively:

Options

G1 Physics · Accepted Answer

Take upward as positive. During ascent, dv/dt = −g − βv. Solve the linear equation: v + g/β = (u+g/β)exp(−βt). At the highest point v=0, so exp(−βtₐ) = g/(g+βu). Therefore tₐ = (1/β)ln(1+βu/g). For height, H = ∫₀→tₐ v dt = ∫₀→tₐ [(u+g/β)exp(−βt) − g/β]dt = u/β − (g/β²)ln(1+βu/g). Hence option B is correct.