Physics

Air resistance acting on a falling body can be taken into account by the approximate relation for the acceleration: a= dv/dt = g - kv, where k is a constant. (a) Derive a formula for the velocity of the body as a function of time assuming it starts from rest (v=0 at t=0). [Hint: Change variables by setting u=g-kv.] (b) Determine an expression for the terminal velocity, which is the maximum value the velocity reaches.

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  1. using their hint, let u = g-kv
    du/dt = -k dv/dt

    -1/k du/dt = u
    du/u = -k dt
    log u = -kt
    u = e^(-kt)
    g-kv = e^(-kt)
    v = (g-e^(-kt))/k
    = g/k - 1/k e^(-kt)

    So, max v is g/k as t->infinity

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