The uncertainty of a toy car's measured average speed

Question

Suppose I have a battery-powered toy car, and I am told to design an experiment which would allow me to determine the average speed of that toy car over, say, a meter, and to account for experimental uncertainty somehow.

I can think of a couple of ways to do this, but no matter which one I consider, it is not clear to me how to incorporate uncertainty. Here is one proposed method:

(1) Put car on floor next to, but 50 cm behind, a meter stick, and turn car on. Get digital stopwatch ready. When front of toy car reaches beginning of meter stick, turn on stopwatch. Turn off stopwatch when front of car reaches end of meter stick. Use distance traveled divided by time to calculate average speed.

Obviously, I could simply get out my calculator and type 100 / (stopwatch reading) and write down the number that I get. However, that does not seem right, for several reasons. The distance (100 cm) must be uncertain because I cannot be confident that the car actually traveled 100.000... cm. The time is also uncertain because it is a digital device, and all digital devices have associated uncertainties, independent of reaction time. It is also uncertain because I cannot be confident that I started and stopped the stopwatch at the moment the car reached the beginning/end of meter stick. I could (somehow) assign an uncertainty to both the distance and time and then use normal rules for assigning an uncertainty of a calculated result. But I cannot think of how to express the value of the uncertainty.

The other method I can think of involves repeating the experiment many times, and then calculating the numerical average and the associated standard deviation. Then, my final answer would be in the form: value +- standard deviation, but it's not clear to me how I would go about rounding wither value; and, rounding those values seem, to me, to be necessary.

How would you experimentally determine the average speed of a toy car using a meter stick and a digital stopwatch only, and incorporate uncertainty of distances, times, and the average speed?

Answer 1

Assuming that the speed of the toy car is constant, which may not always be the case, the uncertainty here will be dominated by your reaction time and your visual perception. I would not worry about the ruler and the stopwatch.

Making a bunch of measurements and averaging the results will reduce your random error. Extending the length of the track will reduce both random and systematic error, since your reaction time will have a smaller contribution to the measured time.

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But, let's say, you want to estimate the uncertainty of the calculated speed based on a single measurement, performed as you've described, and, for that, you need to know the uncertainties of your measurements. One possible approach here is to perform some preliminary tests to "calibrate" yourself, so to speak, since, as mentioned before, your reaction time and visual perception will be dominant factors affecting the uncertainty of your results.

You can estimate your reaction time by timing known intervals with your stopwatch. Here is a reference clock you can use for this purpose. You can make a number of measurements of some interval, say, $3$ seconds, and determine the standard deviation of your measurements, which will give you an idea of uncertainty in the measured time.

To estimate the uncertainty in the perception of distance, you can place the toy car at random locations close to the beginning and then close to the end of the meter stick and try to quickly assess the position of the car from the location(s) you use to perform your speed measurements. By doing it a number of times and then comparing your quick assessments with more carefully measured positions of the car, you'll get an idea of the uncertainty in the measured distance.