Approaching Calculus-Based Introductory Physics

[PhotonTrail]

I'm currently self-studying introductory physics. I am able to understand the derivation of equations in my notes, but I find myself struggling to apply calculus in the problem sets. I tend to instinctively solve the questions algebraically, and will begin to really contemplate using calculus only if the question is too complex to solve algebraically.

Is this the right approach? I have a nagging suspicion that I'm not approaching the questions at the depth I am supposed to. Sure I can get the correct solution, but it feels as if I took a shortcut of sorts. Is it a problem with me, or is it simply a matter of questions that are too simple?

 

[bp_psy]

If it can be done correctly with basic algebra then it should be done with basic algebra. Trying to do it with calculus will only be an exercise in over-complicating stuff. In basic intro mechanics you need calculus to derive your basic equations of motion and you are mostly done.. Doing those simple problems with calculus means that you are just deriving the same formulas over and over and there is very little insight to gain. If you are using a more advanced introductory text like Kleppner/Kolenkow you will have to use calculus because the problem actually requires it, not because you force yourself.

 

[jimgram]

Ask yourself if a problem is linear or non-linear. For example, if car (or particle)is moving and has a force acting on it (say torque from an automobile engine) then it is accelerating and you need to know its velocity after a period of time. The linear kinematic equation is is v_f=a*t. But an internal combustion engine does not have a linear torque curve - it's torque is different at different speeds. So, acceleration is non-linear over time. In this case, v equals the integral of acceleration. You must use calculus. I think you'll find most physics problems are non-linear.

 

[DimReg]

In physics, you have the choice of problem solving technique, as long as it works. The fact that you are often finding simple solutions of these problems is a good thing, because it means you are actually thinking about your solutions. It's actually an important skill to recognize when more advanced math techniques are needed/

If you want more practice using calculus, try generalizing things in your problems, such as making the coefficient of friction depend on position, or make densities not uniform.

 

[PhotonTrail]

Thanks for the replies!

I gathered that it's okay to use algebraic solutions as long if the questions are not demanding enough. But how does one know which algebraic equations to commit to memory? Stuff like $s=ut+\frac{1}{2}at^2$ and $\vec{E}=\frac{\sigma}{2\epsilon_0}$ are trivial enough to remember and come naturally, but where does one stop? Should I memorize equations like the one for the electric field due to a dipole $\vec{E}(\vec{r})=\frac{1}{4\pi\epsilon_0}(-\frac{\vec{p}}{r^3}+\frac{3(\vec{p}\cdot\vec{r}) \vec{r}}{r^5})$ or hope to have enough time to derive them every time I need to use them?

 

[DimReg]

That depends on what you are learning physics for. If it's just for fun/personal enrichment, just memorize only what seems important. If it's for class, memorize what the teacher expects you to memorize (you can ask them). If it's for work, then you'll memorize what you need because you'll use it over and over.

It's more useful to become familiar enough with the fundamentals to derive simple formulas like E = σ/2ϵ0 quickly, than to have that formula memorized. Two good examples are the formulas you cited as "simple" formulas. If you can't derive them, then you don't understand the basics of physics well enough. The dipole formula on the other hand takes more work, and it's reasonable if you don't know how to derive it (I don't know off the top of my head). But you should surely know what a dipole is. Edit: note that since I know what a dipole is, I know where to start rederiving the electric field. At that point, it's just a matter of if I can solve the math.

Remember, it's physics not biology. Memorize the formula for moment of inertial, but don't memorize the moment of inertia of a wheel. Just be economical in your memorization.

 

[BruceW]

The linear kinematic equation is is v_f=a*t. But an internal combustion engine does not have a linear torque curve - it's torque is different at different speeds. So, acceleration is non-linear over time. In this case, v equals the integral of acceleration. You must use calculus. I think you'll find most physics problems are non-linear.

That's not the usual definition of a non-linear system. But I would agree that generally in physics problems, the acceleration will not be constant.

I gathered that it's okay to use algebraic solutions as long if the questions are not demanding enough. But how does one know which algebraic equations to commit to memory?

Yeah, good question. I guess there will always be a mixture of some equations which you have memorised, and some equations which you can derive. When I was undergraduate (not so long ago), a fairly large part of the course was to learn how to do various derivations. At the very least, there is always some reasonable explanation for why a certain equation 'works'.

Also, there are equations which you probably don't really need to memorise or be able to derive. For example, the equation for the dipole field as you mentioned. I don't think I was ever required to memorise it for my undergraduate. And I could probably derive it, given some time, but definitely not in a couple of minutes. The main thing is to understand what the dipole is (as DimReg says), and the important physical principles, e.t.c.
Of course, if you are doing a project, or research, or are particularly interested in dipoles, then you will probably know the equation off by heart. But this will probably happen naturally anyway, if you use an equation enough times, it tends to stick :)