MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A hovercraft travels in a straight channel. Engineers report that “the craft’s acceleration at any moment is directly proportional to the difference between 18 ft/s and its instantaneous velocity.” Assume motion is along a straight horizontal axis and that velocity is positive in the forward direction.
(a) Write a differential equation that models the hovercraft’s velocity v(t) (in ft/s) as a function of time t (in seconds). Clearly define any proportionality constant you introduce.
(b) Describe, without solving the equation, how the sign of the acceleration changes as the craft speeds up or slows down. Explain what this indicates about the long-term behavior of the velocity.
(c) Suppose the craft starts from rest at t=0. Explain why the initial value you choose is appropriate, and discuss whether the model predicts that the craft can exceed 18 ft/s.