Friday, June 15, 2012

Measurements and theory - the required number of data points

I hosted group meeting today. The topic was on Brownian motion, but during it I came to an interesting realization about measurements of displacement, velocity, and acceleration.

Consider the measurement of a displacement of an object, say a car, from some fixed point, let's say its owner's house. A measurement of the car's distance from home consists of taking a yardstick and determining how many yardsticks away from the house the car lies at a moment in time. This is one measurement. If we wish to determine the car's (average) velocity, we measure the distance at another moment in time, subtract this from the previous measurement to obtain its displacement, and divide by the time interval between measurements.

This means that if we wish to determine the velocity of the car, we necessarily must make two measurements of its distance from home. If the car is accelerating, then this may be determined from a minimum of three distance measurements. The two (average) velocities determined by the three points where the distance measurements occurred are subtracted and divided by the time difference between the first and last measurement, resulting in the car's acceleration.

One could argue that the car has an instantaneous velocity and acceleration, so to talk about the number of measurements required to determine one of the car's dynamical quantities is nonsense. For a car this is more or less true, practically speaking. However, the fact is that at very small length and time scales, we can not measure instantaneous quantities. Rather, they are sampled at specific time intervals. For example, a digital camera works by sampling the light that falls on each pixel of its CCD array. So, any measurement is in reality a collection of discrete points of data. Quantum mechanics also tells us that in the microscopic world, measurements are discrete and that the continuum results from combining a large number of discrete measurements.

Back to my main point. In kinematics and calculus we learn that velocity is the first order derivative of displacement and that acceleration is the second order derivative. The previous argument suggests that if we wish to determine some quantity that is a derivative of the quantity that we actually measure, then we need to make N+1 measurements, where N is the order of the derivative representing the quantity that we're after.

So there you have it. You can't determine an acceleration from one or two measurements of displacement. If I had a device that measured velocity, however, then the acceleration could be determined from two velocity measurements, since acceleration is the first derivative of velocity. All of these arguments aren't too surprising when you consider that these quantities are essentially differences, and differences require more than one thing.

As an aside, fractal motion becomes a bit more enlightening in this context, but I'll save that for a later discussion.