Amotorboat starts from rest (initial velocity v(0)=v₀=0). its motor provides a constant acceleration of 4 ft /s², but water resistance causes a deceleration of v²/400 ft /s². find v when t=7 s, and also find the limiting velocity as t → +[infinity] (that is, the maximum possible speed of the boat).

With , we have by the fundamental theorem of calculus

Differentiating both sides wrt gives

This equation is separable as

Integrate both sides; on the left, substitute

Given that , we have

So the velocity at time is that satisfies

When , we have

We can rewrite the particular solution as

Taking the limit as on both sides gives

(the exponential terms approach 0)

so the limiting velocity, call it , satisfies the quadratic equation

Realistically, the boat won't speed up enough for the resistance to be so strong as to reverse the boat's direction, so the limiting velocity should be positive.

is there any photos or whatever?