Three forces act significantly on a freely floating helium-
filled balloon: gravity, air resistance (or drag force), and a
buoyant force. Consider a spherical helium-filled balloon
of radius $r=15 \mathrm{cm}$ rising upward through $0^{\circ} \mathrm{C}$ air,
and $m=2.8 \mathrm{g}$ is the mass of the (deflated) balloon itself.
For all speeds $v,$ except the very slowest ones, the flow of
air past a rising balloon is turbulent, and the drag force $F_{\mathrm{D}}$
is given by the relation
$$F_{\mathrm{D}}=\frac{1}{2} C_{\mathrm{D}} \rho_{\mathrm{air}} \pi r^{2} v^{2}$$
where the constant $C_{D}=0.47$ is the "drag coefficient"
for a smooth sphere of radius $r .$ If this balloon is released
from rest, it will accelerate very quickly $($ in a few tenths
of a second) to its terminal velocity $v_{T},$ where the
buoyant force is cancelled by the drag force and the
balloon's total weight. Assuming the balloon's accelera-
tion takes place over a negligible time and distance, how
long does it take the released balloon to rise a distance
$h=12 \mathrm{m} ?$
