Question

Use the force balance Fd = Fg −Fb to derive a formula for the terminal settling (fall) speed vs of the particle. Simplify as far as possible (Hint: replace the particle mass with its density and volume, the latter then with a radius based description).

Answer 1

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 (vT). Since Fd is proportional to the speed squared, a heavier skydiver must go faster for Fd to equal his weight. Let’s see how this works out more quantitatively.

At the terminal velocity,

Fne t= mg−Fd = ma = 0

Thus,

mg=Fd

Using the equation for drag force, we have

mg=12CρAv2T

Solving for the velocity, we obtain

vT=√2mgρCA

Assume the density of air is ρ=1.21kg/m3. A 75-kg skydiver descending head first has a cross-sectional area of approximately A=0.18m^2 and a drag coefficient of approximately C=0.70 . We find that

vT=√2(75kg)(9.80m/s2)(1.21kg/m3)(0.70)(0.18m2)=98m/s=350km/h

This means a skydiver with a mass of 75 kg achieves a terminal velocity of about 350 km/h while traveling in a pike (head first) 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.