Answer
a) $v=\frac{mg}{b}(1-e^{\frac{-bv}{m}})$
b) The proof is below.
Work Step by Step
a) We know that gravity is causing the object to speed up, and we know that the drag force is causing the object to slow down. Thus, we know that the acceleration is:
$ma= mg - bv \\ a = g-\frac{bv}{m}$
Thus, we find that $\frac{-bv}{m}$ is what the natural exponential will be raised to. We know that as time approaches infinity, the object approaches terminal velocity. Thus, we find the expression for v:
$v=\frac{mg}{b}(1-e^{\frac{-bv}{m}})$
Note, the coefficient of this equation is $\frac{mg}{b}$, for this is the terminal velocity that the book tells us.
b) We are asked to prove the terminal velocity. We know that it occurs when the force of gravity and the force of drag are equal. Thus:
$mg=bv \\ v =\frac{mg}{b}$
