(III) The force of air resistance (drag force) on a rapidly

falling body such as a skydiver has the form $F_{\mathrm{D}}=-k v^{2},$ so

that Newton's second law applied to such an object is

$$m \frac{d v}{d t}=m g-k v^{2},$$

where the downward direction is taken to be positive.

(a) Use numerical integration to estimate (within 2$\% )$ the

position, speed, and acceleraton, from $t=0$ up to

$t=15.0 \mathrm{s},$ for a $75-\mathrm{kg}$ skydiver who starts from rest,

assuming $k=0.22 \mathrm{kg} / \mathrm{m} .$ (b) Show that the diver eventually

reaches a steady speed, the terminal speed, and explain why

this happens. (c) How long does it take for the skydiver to

reach 99.5$\%$ of the terminal speed?