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# 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?

## 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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