Terminal Velocity Some interesting situations connected to Newton's second law occur when considering the effects of drag forces upon a moving object. For instance, consider a skydiver falling through air under the influence of gravity. The two forces acting on him are the force of gravity and the drag force (ignoring the small buoyant force). The downward force of gravity remains constant regardless of the velocity at which the person is moving. However, as the person's velocity increases, the magnitude of the drag force increases until the magnitude of the drag force is equal to the gravitational force, thus producing a net force of zero. A zero net force means that there is no acceleration, as shown by Newton's second law. At this point, the person's velocity remains constant and we say that the person has reached his terminal velocity (vr) . Since Fp is proportional to the speed squared, a heavier skydiver must go faster for Fp to equal his weight. Let's see how this works out more quantitatively. At the terminal velocity, Fuet=mg- Fp=ma = 0. Thus, mg = FD. Using the equation for drag force, we have mg =CpAv}. Solving for the velocity, we obtain 2mg VT = V PCA Assume the density of air is p = 1.21 kg/m3. A 75-kg skydiver descending head first has a cross-sectional area of approximately A = 0.18 m2 and a drag coefficient of approximately C = 0.70. We find that = 98 m/s = 350km/h. (1.21 kg/m3) (0.70) (0.18 m2) 2 (75 kg) (9.80 m/s2) This means a skydiver with a mass of 75 kg achieves a terminal velocity of about 350 km/h while traveling in a headfirst position, minimizing the area and his drag. In a spread-eagle position, that terminal velocity may decrease to about 200 km/h as the area increases. This terminal velocity becomes much smaller after the parachute opens.
Terminal Velocity of a Skydiver Find the terminal velocity of an 85-kg skydiver falling in a spread-eagle position. Strategy At terminal velocity, Fnet = 0. Thus, the drag force on the skydiver must equal the force of gravity (the person's weight). Using the equation of drag force, we find mg = {PCAv2. Solution The terminal velocity UT can be written as 2mg VT = > pCA 1 = 44 m/s. (1.21 kg/m3) (1.0) (0.70 m2) 2 (85 kg) (9.80 m/s2) Significance This result is consistent with the value for Vy mentioned earlier. The 75-kg skydiver going feet first had a terminal velocity of UT = 98 m/s. He weighed less but had a smaller frontal area and so a smaller drag due to the air. Check Your Understanding Find the terminal velocity of a 50-kg skydiver falling in spread-eagle fashion. 34 m/s The size of the object that is falling through air presents another interesting application of air drag. If you fall from a 5-m-high branch of a tree, you will likely get hurt-possibly fracturing a bone. However, a small squirrel does this all the time, without