The force of air resistance on a sphere of radius $R$ can plausibly be argued to have the form
$$
\vec{F} \text { drag }=-\frac{1}{2} C \rho R^{2}|v| \vec{v}=-b|v| \vec{v}
$$
where $\vec{v}$ is the vector velocity and $|\vec{v}|$ is its magnitude (the speed). The density of the air, $\rho$, is about $1 \mathrm{~kg} / \mathrm{m}^{3}-1 / 1000$ that of water. The parameter $C$ is a dimensionless constant.
If we drop a steel ball and a styrofoam ball from a height of $s$. the steel ball reaches the ground when the styrofoam ball is still a bit above the ground. Call this distance $h .$ Estimate the air resistance coefficient $C$ as follows:
(a) Assume the effect of air resistance on the steel sphere is negligible. Calculate approximately how long the steel sphere takes to fall to the ground $\left(\Delta t_{\text {ste }}\right)$ and how fast it is traveling just before it hits $\left(v_{\text {ste }}\right) .$ Express your answers in terms of $s, g$, and $m$.
(b) Since the steel and styrofoam were not very different, use $\left\langle\vec{v}_{\text {ste }}\right.$, the average velocity of the steel ball during its fall to calculate an average air resistance force, $(\vec{F}$ drag $)=-b\left\langle\left.\vec{v}\right|^{2}\right.$ acting on the styrofoam sphere during its fall. Express this force in terms of $b, m$ (the mass of the styrofoam sphere), $g, s$, and $h$.
(c) The average velocity of the steel ball is $\left\langle\vec{v}_{\text {ste }}\right\rangle=s / \Delta t_{\text {ste }} .$ The average velocity of the styrofoam sphere was $\left\langle\vec{v}_{\text {sty }}\right\rangle=(s-h) / \Delta t_{\text {ste }}$ Assume this difference, $\Delta(\vec{v})$, is caused by the average air resistance force acting over the time $\Delta t_{\text {ste }}$ with our basic Newton's law formula:
$$
\left\langle\vec{F}^{\text {drag }}\right\rangle \Delta t_{\text {ste }}=m \Delta(\vec{v}) .
$$
Use this to show that
$$
b \cong \frac{m h}{s^{2}}
$$
(d) A styrofoam ball of radius $R=5 \mathrm{~cm}$ and mass $m=50 \mathrm{~g}$ is dropped with a steel ball from a height of $s=2 \mathrm{~m}$. When the steel ball hits, the styrofoam is about $h=10 \mathrm{~cm}$ above the ground. Calculate $b$ (for the styrofoam sphere) and $C$ (for any sphere).