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The Real World: Air Resistance

Michael Fowler

But wasn't Aristotle Right Sometimes?

We've already admitted that for a ball falling through water Aristotle's description is closer to the truth than Galileo's "naturally accelerated motion". Galileo was well aware of that, and his response was that the true natural motion could only take place in a vacuum, otherwise the medium the body was falling through would impede the acceleration to some degree. Galileo's insight was that for a reasonably heavy body falling a few meters through air, the air resistance didn't make a big difference. He did state explicitly, though, that even for heavy materials, air resistance was important at the speeds attained by firearms (see the last three paragraphs of thisóbut can you spot a mistake in the third paragraph from the end? We'll discuss this in the next paragraph below).

Terminal Velocity

In practice, as Galileo understood, the air resistance to a falling object increases with speed. But he did not think as we do in terms of forces, and the force of gravity, the weight, pulling the ball down. So he thought that since a ball of wood was more slowed in its fall by air resistance than a ball of lead of identical size and shape, the ball of wood must be experiencing a greater retarding force. We now understand that the drag force is the same, but since the lead weighs more, the same drag force is relatively less important to the heavier ball. A full discussion, though, needs Newton's Laws of Motion, which we'll be coming to later.

Video Check of Terminal Velocity

It is relatively easy to see the transition from constant acceleration to terminal velocity using the video camera. We use those large coffee filters with flat bottoms and fluted edges. These can be stacked inside each other, so we can be confident that the drag force on one filter as it falls is almost exactly the same as the drag force on eight filters fitted snugly together. We found in a quick experiment that a single filter reached a terminal speed of about one meter per second, four filters packed together reached two meters per second. Thinking ahead a little bit, if the drag force balances the weight a steady velocity, the drag force is evidently proportional to the square of the speed. It is instructive to plot, using Excel, the velocity/time curves for different numbers of filters.

Actually, the dependence of drag force on speed is quite a complicated subject. For a ball falling slowly through a viscous fluid, the force is proportional to the speed. Imagine a steel ball falling through molasses. This kind of smooth flow leaves no "wake", there is no turbulence. But even the coffee filters falling through air leave a trail of disturbance, and under these circumstances it turns out that the drag force increases as the square of the speed.