Chapter 3.3, Problem 83ES

Suppose that a steel ball bearing is released within a vat of fluid and begins to sink. According to one model, the speed $\upsilon \left(t\right)$ (in m/s) of the ball bearing $t$ seconds after its release is given by the formula

$\upsilon \left(t\right)=\frac{9.8}{k}\left(1-e^{-kt}\right)$

where $k$ is a positive constant that corresponds to the resistance the fluid offers against the motion of the bearing. (The smaller the value of $k$ , the weaker will be the resistance.) For $t$ fixed, determine the limiting value of the speed as $k\to 0^{+}$ , and give a physical interpretation of the limit.