A Bar Trick

In a certain lucrative bar trick, a naive bystander is asked to try to drop a playing card into a hat at his feet. The idea is demonstrated to him by the crafty challenger, who holds the card vertically and drops it over the hat. Each time the bystander drops the card in this manner, it flutters wildly in some apparently random direction and completely misses the hat. The challenger bets that he can hit the hat with just one drop. Bets are placed. The challenger then steps up and drops the card with a horizontal initial orientation, and it falls obediently into the hat. This simple trick illustrates several fundamental aspects of the dynamics of a falling thin plate. First, as shown by the behavior of the card when dropped vertically, the motion of such an object can be very chaotic. Here, one of the key features of chaotic dynamics---a severe sensitivity to the precise initial conditions---is what makes the scheme unprofitable for the bystander. Second, the severity of this dependence can itself be a function of the value of the initial condition. Thus, a card dropped with an initially horizontal orientation remains nearly so; for vertical distances typical for this trick, initially different trajectories diverge only slightly, reaping the challenger his profit.

Photo from The University Record (July 30, 97)

Understanding the motion of bluff objects falling in a viscous medium is of importance, and has applications in many disciplines, including meteorology, sedimentology, aerospace engineering, fluid dynamics, and chemical engineering. The study of this problem goes back to at least Newton, who observed complex motion of objects falling in both air and water. The problem was also studied by Maxwell, who discussed the motion of a falling strip of paper in a beautiful 1854 article. Kelvin, Stokes, Kirchhoff, and many others have studied this problem. More information on it can be found in the postcript (PS) and PDF versions.