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Problem 50 Easy Difficulty

An object of mass $ m $ is moving horizontally through a medium which resists the motion with a force that is a function of the velocity; that is,
$ m \frac {d^2s}{dt^2} = m \frac {dv}{dt} = f(v) $
where $ v = v(t) $ and $ s = s(t) $ represent the velocity and position of the object at time $ t, $ respectively. For example, think of a boat moving through the water.
(a) Suppose that the resisting force is proportional to the velocity, that is $ f(v) = -kv, k $ a positive constant. (This model is appropriate for small values of $ v. $) Let $ v(0) = v_0 $ and $ s $ at any time $ t. $ What is the total distance that the object travels from time $ t = 0? $
(b) For larger values of $ v $ a better model is obtained by supposing that the resisting force is proportional to the square of the velocity, that is, $ f(v) = kv^2, k > 0. $ (This model was first proposed by Newton.) Let $ v_0 $ and $ s_0 $ be the initial values of $ v $ and $ s. $ Determine $ v $ and $ s $ at any time $ t. $ What is the total distance that the object travels in this case?

Answer

a) Terminal velocity is $\frac{g}{k}$
b) $\frac{m v_{0}}{k v_{0} t+m}, \frac{m}{k} \ln \left(\frac{k v_{0} t}{m}+1\right)+s_{0},$ infinite

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