For objects with linear size between a few millimeters and a few meters moving through air near the ground, and with speed less than a few hundred meters per second, the drag force is close to a quadratic function of velocity, \(F_{D}=(1 / 2) C_{D} A ho v^{2}\), where \(ho\) is the mass density of air near the ground, \(A\) is the cross-sectional area of the object, and \(C_{D}\) is the drag coefficient, which depends upon the shape of the object. A rule of thumb is that in air near the ground (where \(ho=1.2 \mathrm{~kg} / \mathrm{m}^{3}\) ), then \(F_{D} \simeq \frac{1}{4} A v^{2}\).

(a) Estimate the terminal velocity \(v_{T}\) of a skydiver of mass \(m\) and cross-sectional area \(A\).

(b) Find \(v_{T}\) for a skydiver with \(A=0.75 \mathrm{~m}^{2}\) and mass \(75 \mathrm{~kg}\). (The result is large, but a few lucky people have survived a fall without a parachute. An example is 21-year old Nicholas Alkemade, a British Royal Air Force tail gunner during World War II. On March 24, 1944 his plane caught fire over Germany and his parachute was destroyed. He had the choice of burning to death or jumping out. He jumped and fell about \(6 \mathrm{~km}\), slowed at the end by falling though pine trees and landing in soft snow, ending up with nothing but a sprained leg. He was captured by the Gestapo, who at first did not believe his story, but when they found his plane they changed their minds. He was imprisoned, and at the end of the war set free, with a certificate signed by the Germans corroborating his story.)