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,

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) = €“ kt, a positive constant. (This model is appropriate for small values of v.) Let v (0) = vo and s (0) = s0 be the initial values of v and s. Determine v and at any time t. What is the total distance that the object travels from time t = 0?
(b) For larger values of a better model is obtained by supposing that the resisting force is proportional to the square of the velocity, that is, f (v) €“ kv2, k > 0. (This model was first proposed by Newton.) Let v0 and s0 be the initial values of and s. Determine and at any time t. What is the total distance that the object travels in this case?

SzP dv dt - f(e) dr