An object of mass m is dropped at time t = 0 and falls under the influence of gravity. Assume that the force o?
of air resistance is proportional to the velocity, and that the acceleration of
gravity is a constant g = 32 f t/sec2
.
(a) Set up and solve the initial value problem to determine the velocity v(t) as a function of
t.
(b) If the terminal velocity is −16 f t/sec, what is the velocity at time t = ln 2?
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(a)
Taking the upward direction as positive,
mv' = -γv - mg
v' + (γ/m)v = -g
φ = e^[∫ (γ/m) dt]
φ = e^(γt/m)
[v e^(γt/m)]' = -g e^(γt/m)
v e^(γt/m) = - ∫ g e^(γt/m) dt
v e^(γt/m) = -(mg/γ) e^(γt/m) + C
v = Ce^(-γt/m) - mg/γ
0 = C - mg/γ
C = mg/γ
v = (mg/γ)[e^(-γt/m) - 1]
(b)
-16 = -mg/γ
v = 16[e^(-gt/16) - 1]
v = 16[e^(ln{2^(-g/16)} - 1]
v = 16[2^(-g/16) - 1]
v = 16[2^(-32/16) - 1]
v = 16[1/4 - 1]
v = -12 ft/s