Putting non-Calculus Physics problems in Calculus form

[zk3381]

I'm a sophomore Physics major currently taking Mechanics, and I recently noticed something when I was going over some homework. I am really good at Calculus, and I noticed I tend to do way better on the calculus based problems (i.e. work, finding force from potential energy etc) than some of the problems that don't involve calculus. So that got me thinking, has anyone ever had the idea of taking information in problems that don't necessarily require calculus to solve them and putting them in some form where you can use calculus (which, for people like me, would make them much easier)? Anyway, I had this thought and wanted to know if anyone else thought of the same thing and what they did.

 

[Overt]

Knowing both ways is helpful. I think knowing how the non-calculus methods are derived from the calculus methods are absolutely beneficial. In general knowing how the less complex methods are derived from the more complex methods is an incredibly valuable and necessary skill. It's one that can and should be learned.

I would say if you're more comfortable with the calculus form of what you're working with then derive how the non-calculus form comes from what you're used to. It will help you in the long run!

 

[PWiz]

I actually feel the same as you. Anywhere you find the word "average", you should immediately know that you can use a derivative or integral over there. As a general rule, replace the ##t## with ##dt## and s with ##ds## (distance) while adding d's to the non-constant variables and you'll convert a standard equation to the calculus form. For example, certain introductory books say that the average force exerted on a collision is given by ##F=m\frac {Δv}{Δt}##. Now ##m## is a constant, and if add the d's to the non-constants, I get ##F=m\frac{dv}{dt}##, which is the calculus form of the equation (the example is too simplistic, but you can apply it to more complex equations using the fact that ##d(a+b)=da +db## and ##d(ab)=da×db##). This process essentially applies to almost all kinematic and dynamic equation and is invaluable too in removing any "approximation" in your answer (although most mechanics problems are mathematical models which "approximately" represent a real life scenario).

 

[zk3381]

I guess what I'm saying is if there is any way to take the information you are given in a general physics problem and turn it into a function you can differentiate or integrate (if that is applicable for the situation)? I mean, in the questions that do have functions in them, they had to get them somewhere (I doubt they are arbitrary). Does that make sense to anyone?

 

[PWiz]

I guess what I'm saying is if there is any way to take the information you are given in a general physics problem and turn it into a function you can differentiate or integrate (if that is applicable for the situation)? I mean, in the questions that do have functions in them, they had to get them somewhere (I doubt they are arbitrary). Does that make sense to anyone?

Are you talking about how to find a function which models real data? If yes, then there are multiple software such as Microsoft Excel which enable you to construct a function which "fits" the given data. By taking Excel as an example, you enter the relevant data onto the spreadsheet, highlight it and make a graph. You can then add a trend line, and Excel will automatically produce an equation which fits the data as much as possible (you can select if you want it to be logarithmic, exponential or a polynomial of a particular degree [higher degrees allow the equation to represent the data better]). In fact, this is a complete branch of mathematics called function reconstruction, and you can be sure that the guys who gave you the equations representing data trends when there were no computer software of this sort really racked their brains

 

[zk3381]

Yeah essentially. They say give you a question on, for example, angular motion and they tell you that the initial angular velocity is x rads/s and it is accelerating/decelerating at a constant whatever rad/s^2. I'm interested in how to take that information and turn it into a function, say (obviously this is wrong but its just an example) y=omega*t+alpha*t^2 or whatever. Obviously knowing how to use kinematics and whatever they tell you in the textbook is the most important thing, but this is more something that I'm just interested in during my spare time (not that I have much of that).