Is the air resistance force on a falling object proportional to its velocity, or to its velocity squared?
When an object falls in air, the air exerts an upward air resistance force (sometimes called a "drag force" ) on it. As the speed of the falling object increases, so does the air resistance force. When the air resistance force equals the weight of the object, it stops accelerating and falls with a constant terminal velocity. Light objects - feathers, pieces of paper, and coffee filters exhibit this behavior quite readily - since they have a large surface area and a small weight, they reach their terminal velocity very quickly.
Certainly, the air resistance force on an object depends on its speed - but how? Suppose that the air resistance force is proportional to the object's velocity. Then, at terminal velocity, vt, the net force on the falling object is zero, so the upward forces and downward forces must balance:
Fair = k vt = W
where k is a constant that depends on the shape of the object and the density of the air, or:
(1)
This equation says that a graph of terminal velocity versus weight will be a straight line with slope 1/k.
On the other hand, what if the air resistance force is proportional to the square of the speed? In this case, at terminal velocity:
Fair = kvt2 = W
(where k is a constant) or:
(2)
which says that a graph of v2 versus W should be a straight line.
| several coffee filters | Pasco motion detector | Pasco computer interface |
First of all, notice that two coffee filters placed together have twice the weight as one coffee filter, but about the same surface area. Three coffee filters have three times the weight as one coffee filter, etc. This means that the appropriate unit of weight in this lab would be "coffee filters", as in "1 coffee filter", "2 coffee filters", etc.
You could place a motion detector on the floor, oriented vertically, and drop coffee filters on it. By analyzing the position vs. time graph of the motion, you should be able to determine the terminal velocity for the dropped coffee filters. After collecting sufficient data, you could plot graphs of v versus W and v2 versus W to determine which (if either) is a straight line.
Which model for air resistance (if any) does your experiment support? Why do you think so? How confident can you be in your conclusion? Discuss the major sources of error in your experiment.