Re: Finding the velocity with non-constant acceleration w/o calculus

The question: "I am teaching a high-school physics class and want to do a lab on terminal velocity. I am dropping aluminum foil objects and can find the average velocity but I want to know the final velocity so I can determine the terminal velocity and the terminal velocity constant for the shape I am using. I am approximating by taking twice the average velocity. Is there a better way?

Some details: I am dropping aluminum foil squares from about 15-20 feet and I think they will reach terminal velocity before they hit the ground. This is a low-tech lab. We have meter sticks and stop watches. We will take several times as the object falls so we should be able to calculate the terminal velocity if the velocity is constant in two consecutive time periods. From there I believe
B = mg / V_t².
Before the object reaches V_t we can find V_avg = d / t.
But V_f = 2 V_avg for V_i = 0 is only strictly true for constant acceleration which is not the case here. I was wondering if there was a simple way to approximate the V_f in this case. Thanks!"

I also teach high-school Physics so I am certainly familiar with the issues at hand! And you are certainly correct that in the case of non-constant acceleration the relation V_f = 2 V_avg does not hold.

In cases like this one of the best ways to determine the item of interest is to

1. take readings on multiple drops of the square,
2. graph distance versus time in a spreadsheet such as Microsoft Excel, and
3. use the spreadsheet's "trend line" capability to determine the terminal velocity from the asymptotic slope of the graph near the end of the drop.

I find that the single most important aspect of this "method" is the multiple trials so that the imprecisions and errors can be reduced by the averaging that is inherent in the spreadsheet's fitted line.

For this experiment the fitted line will need to be something other than linear, with a fourth-order polynomial or an exponential fit probably suitable. (I'm sure you know that in Excel the equation of the fitted line can be displayed on the graph.)

John Link, MadSci Physicist