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The trouble Im having is with instantaneous velocity or acceleration. Almost all questions give a function that I can take a limit of. But how would one model data as a function? Oh and It cant be linear..

Andrew Graham
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    It's an idealization. Typically, the time-scale over which the quantity measured is finite, but, so small that for all intents and purposes the instantaneous change is a better description than say the technically correct average. – James S. Cook Nov 27 '13 at 02:48

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If you have a number of data points, you can fit a function to them. There are many methods of doing this. However, before you even start, you should have some idea of what kind of function represents the phenomena.

For example, if you are dropping a ball from a rooftop and want to model its velocity and acceleration on the way down, you can measure the distance at various time points. You can then fit a quadratic curve to your data, because you know the distance is a quadratic function of time for problems of this sort. Once you have decided on a quadratic curve, you can take derivatives to get velocity and acceleration.

Of course, your answers are only as good as your measurements. They may also vary considerably depending on how you chose to fit the curve. For example, a least squares fit will give you a different answer than a max fit.

So how do you know what kind of function to pick, and what sort of fit to use? It depends on the situation. Sometimes a lot is known and/or there are ways of checking your error; sometimes not much. In the latter case, you have to do lots of experiments and try to come up with a good model. It's not easy. You could take a whole course in that subject, and then need to build up lots of experience to get good at it.

Betty Mock
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  • To make the second paragraph specific, the height would be modeled as $h(t)=h(0)-\frac 12gt^2$ Since everything else is known, you could use this to measure the height of a building by dropping something (preferably dense) off it and measuring the time of fall. The next more accurate model is to incorporate air resistance. Now you have $h''=-g+k(h')^2$, where the fact that drag goes as $v^2$ is correct but not obvious. You can now use your data to fit and find $k$ – Ross Millikan Nov 27 '13 at 05:33