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) = −kv, k a positive constant. (This model is appropriate for small values of v.) Let v(0) = v0 and s(0) = s0 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 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) = −kv2, k > 0. (This model was first proposed by Newton.) Let v0 and s0 be the initial values of v and s. Determine v and s at any time 1. What is the total distance that the object travels in this case?
Want to see the full answer?
Check out a sample textbook solutionChapter 9 Solutions
Calculus: Early Transcendentals
- College Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage Learning