Aerodynamics is important for designing aircraft, fuel-efficient cars, and parachutes, among many other things. An important aspect is that it takes work to push air out of the way of a moving object.
In our vertical wind tunnel, 50 cm in diameter, we examine the balance between downward gravitational force and the upward aerodynamic drag force. People can study how fast objects of different sizes and shapes fall with and without wind and can even find the wind speed where an object levitates—neither rests on the bottom nor climbs out of the tunnel.
Newton’s law of gravity describes how different objects attract each other. The force pulling any object toward earth, which we call the gravitational force, is proportional to its mass times the acceleration of gravity, g:
Fg = mass × g
The value of g is 9.8 m/s2 = 32 ft/s2 = 22 miles/hr/s. Multiplying acceleration by time gives the velocity.
The last units are useful to discuss acceleration of cars. Sports cars are often characterized by the number of seconds it takes to get to 60 miles/hr. A Ferrari LaFerrari, Porsche 911 Turbo S, and some Tesla models can reach 60 miles/hr in about 3 seconds. That means the average acceleration is about the same as for gravity.
In a vacuum, all objects pick up speed at the same rate, and the speed (velocity) attained is proportional to time almost without limit. But in air, the effort needed to push the air out of the way increases as the square of the velocity.
Fd = density of the air × velocity squared × cross sectional area of the object × drag coefficient
That is why gas mileage in a car decreases when driving 80 miles/hr compared to 55 miles/hr. The drag coefficient depends on the shape of the objects, and a boxy van has a drag coefficient 2-3 times larger than a sporty car.
A falling object will reach what is called the terminal velocity, when the force of gravity equals the force needed to move the air out of the way, that is Fg = Fd. A parachute has a big cross sectional area to bring heavy objects gently back to earth. Since the drag is proportional to air density, it would take a much larger parachute to land safely on Mars.
All these facts are summarized in the following figure. At terminal velocity, the blue and green arrows are equal.
With this background, we analyze some data taken in our wind tunnel for three polystyrene foam balls (2.5 to 4.9 cm) and the 9-cm and 18-cm diameter beach balls shown in the first picture (all spheres with drag coefficients of 0.47). We measured the wind velocity with an anemometer, with measures the rate at which a small propeller spins in the wind. The wind velocity varies somewhat across the diameter of the tunnel, which causes the spheres to move up and down as they move around, so we used the average wind velocity. We obtained good agreement with the equation in the preceding figure, considering the limitations of our apparatus. One challenge to think about is why the levitation velocity of the foam balls changes with diameter, while the beach balls both have about the same levitation velocity.