# Algebra based -> calculus based physics

hey guys, I am currently in high school and I finished a college-level class in algebra-based physics, I'm going to independently study calculus based physics next year. What is the difference between algebra based physics and calculus based physics? (ie. Newtonian mechanics in an algebra based class and Newtonian mechanics in a calculus based class etc.) I want to know these differences so I can start from there. I already have the basic conceptual knowledge. As for calculus, I've learned limits, derivatives, indefinite integrals, and definite integrals. I have the textbook Physics: Calculus by Eugene Hecht and I'm not sure exactly how to use it because it teaches the concepts of physics from the beginning again, which is frustrating because I just want to start where the calculus comes in. I have access to other physics textbooks from a physics library so you guys can suggest to me a more suitable text if you have one in mind. All help will be greatly appreciated.

Well, at your level I'm thinking the only difference is you'd learn concepts such as non-constant forces, non-constant accelerated motion (i.e. s=s(t), v=ds/dt, etc.), and "conservative forces", i.e. F = del V.
Also, you could expect rigid body dynamics to come into play (maybe even polar or cylindrical co-ordinates), with moment of inertia calculations to come into play. (Really fun stuff, if you ask me)
That's just Newtonian mechanics though.
With electromagnetism, depending on what level that text is at, you'll probably deal with stuff like "Gauss' Law" and making use of symmetry and uniform charge distributions to determine electric fields and forces on charges and the like. Maxwell's Equations would come in if this text is daring enough to deal with vector calculus.
And then there's thermodynamics-- you'll probably see quite a bit of partial derivatives there.

With respect to your frustration at the repitition of already known information, you might want to go over that again because, now that the reader is assumed to have a working knowledge of calculus, the connection between physics and calculus would become a lot clearer.
Then again, you might already know how to apply a "limiting process" to your already developed stuff to connect the dots. :)
Have fun with that. You might be interested in taking the AP Physics C exam if you're motivated enough. (It's not really challenging or too reflective on a REAL college physics course-- but it'd be proof enough for whatever college you want to go to)

Integral
Staff Emeritus
Gold Member
Trouble is, Calculus based Physics does not start at some point after Algebra based Physics. They cover the same material, just use different math. In Algebra based Physics you will have to memorize equations like:

$$x = \frac {at^2} 2 + v_0 t + x_0$$

In calculus based physics you learn that this equation is just the integral of

$$\ddot x = a$$

So you will be seeing a lot of repeat physics, just the math will be different and you will learn were many memorized equations come from.

I took a college algebra-based physics class in high school, followed by calculus-based physics my freshman year of college. My first semester, which was freshman classical mechanics, the professor and the book presented concepts involving calculus now and then. But not once did I have to actually do calculus on any exam problem. And the whole semester there were only two homework problems (which we didn't have to turn in anyway) which required any knowledge of calculus. Basically the only difference was that the problems we were given were much more difficult. There were more forces to deal with, more effects to consider, etc.

My second semester on freshman electricity and magnetism relied quite heavily on calculus. Finding electric and magnetic fields required heavy use of integrals. Maxwell equations such as the integral form of Gauss' Law and Ampère's Law were also covered in-depth. So calculus became indispensible at this level. I think the reason for the discrepancy between physics 1 and 2 is that you're expected to take calculus concurrently with physics. So they don't expect you to know very much calculus your first semester. Your second semester, you've already learned about derivatives and integrals, and so they'll expect you to solve problems involving these concepts.

Anyway, that was my personal experience when I was in college. Maybe other schools are different. Indeed, the school where I'm currently doing my graduate studies has a somewhat different set of freshman undergraduate physics classes (they cover the same material in two semesters that I did in three, meaning it's probably simplified quite a bit). So what I said might not fully apply to your calculus-based physics class.

mathwonk
Homework Helper
is the question only about passing courses? or acquiring understanding? you cannot possibly grasp physics without knowing calculus it seems to me.

I would counter mathwonk's claim that you must have calculus to understand physics; at least in the beginning. I teach an algebra based physics class at my university, and although some of the concepts will seem disconnected- it is possible to teach and learn nearly every concept from a fundemental physics course (a college freshmen class), without the aid of calculus. The only real difference will be that the number of problems one can address without the use of calculus is limited, or that solving some introductory problems require a 'faking' of the calculus (newton's second law in the form of dP/dt=F, for instance).

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As for ct1220's question:

You already have had your introduction, you don't necessarily need to do another introductory physics course again, this time with calculus. You could, I did. Heck, I took both AP Physics B and AP Physics C (both sections), and still decided to take Physics for engineers and Scientists (which was just another fundemental physics class). If you choose not to review the physics with a new mathematical method, you might want to look into picking up a classical mechanics textbook (Something like Analytical Mechanics by Fowles and Cassiday) and tinker around in there, as that is the next step from a fundementals class. The classical mechanics books will likely be a difficult jump since it sounds like you just finished a year of calculus and may not be as confortable with it as the authors expect; however, if you are looking for a more indepth view of physics, there really is no other way.

Thanks for all the replies guys, they are really helpful. It seems it would actually be a good idea for me to go through the fundamentals again, but with calculus (the Analytical Mechanics book seems like too big of a jump). Exactly how much calculus would be used for say.. the AP Physics C exam?

If you have seen the syallibus/suggested topics for the AP Calculus BC test (which is pretty much calc 1-2, infinite series, polar coordinates (maybe), and primative vector calc), I would suggest it. Many of the math gurreu's will give you a hard time since AP calc has very little in relation to a true college math class; however, you are just going to want a familarity and a very very rudementry understanding of the topics in the AP Calc. BC realm to pass the AP Physics C test, at least if you plan on doing both sections.

When I was in high school I did an independent study of AP Calc BC and AP Physics C, and I concentrated primarily on the E&M sections of the Physics C because it had the "most calculus" between the two sections.

The AP physics C test is likely going to be a poor representation of college physics, but hey your in high school and any practice with this material is going to help.

mathwonk
Homework Helper
come on. if physics could be done without calculus, then newton would not have invented it. i rest my case.

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Exactly how much calculus would be used for say.. the AP Physics C exam?
Not much-- basic derivatives and integrals.
Atleast, for the mechanics part. (I didn't do the E&M one)
The only real tricky part (which they haven't included since the '80s I think) is setting up your integrals-- i.e. when calculating moments of inertia, especially if you don't know double/triple integrals.
But don't stress over the calculus-- I took the course only after taking a basic self-study calculus course and did well.
If you understand the physics, you'll ace it.

come on. if physics could be done without calculus, then newton would not have invented it. i rest my case.
True. Then again, someone might further argue, "but Newton chose to create calculus to formulate his version of mechanics."
However, to counter that, why have other formulations of mechanics (Lagrangian & Hamiltonian) relied on calculus?
I agree with ^_^physicist though-- for the very basics, not much calculus is required.

come on. if physics could be done without calculus, then newton would not have invented it. i rest my case.
Lets not start the who invented calculus fight; it can it ugly. However, I will state that Newton was continously irriated by his original "calculus." Rather Newton wanted to demonstrate that mechanics could be described by geometric arguements.

An excellent book discussing this topic is Niccolo Guicciardini's Reading the Principia .

That is beside the point, however, as I do agree with you mathwonk in that to reach the nitty-gritty of physics you must have a mastery of the techniques of calculus.
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The only real tricky part (which they haven't included since the '80s I think) is setting up your integrals-- i.e. when calculating moments of inertia, especially if you don't know double/triple integrals.
When I took the AP Physics C test, I believe that this was excluded from the test, or it was miniumal. However, I did need to have an understanding of surface integrals for some problems in the E&M section of the test.

come on. if physics could be done without calculus, then newton would not have invented it. i rest my case.
The Principia is physics without calculus.

Thanks for all the replies guys, they are really helpful. It seems it would actually be a good idea for me to go through the fundamentals again, but with calculus (the Analytical Mechanics book seems like too big of a jump). Exactly how much calculus would be used for say.. the AP Physics C exam?
seems to me like you have an adequate calculus background. it's been my experience that there's much more actual calculus involved in the problems relating to EM than basic classical mechanics.

no idea if you do "shell law" problems in physics c, as i only took physics b in high school and then the honors introductory physics sequence at my college.

(fun fact: i taught myself (the ideas of) calculus using "the complete idiot's guide to calculus" just so i could get more out of the physics for engineers and physical scientists book my brother let me have after he unsuccessfully tried to sell it at a reasonable price to the university bookstore.)

hi.
Giancoli Physics for Scientists and Engineers with Modern Physics (3rd Edition), for the calculus-based General Physics course primarily taken by engineers and science majors (including physics majors). Physics for Scientists and Engineers, Pt. 1 (Third Edition) is part 1. One review says "...more problem solving (than Serways's text), detailed proofs are included." Another review says "...uses only basic calculus". Another review says "It abounds with great number of examples and problems." and "This book is currently adpoted as the textook at UC-Berkeley (Physics 7 series) and MIT (physics 8.01)." Another reviewer says this book is used by the physics 4A-E series at UCSD. Another reviewer thinks it relies too much on special cases. Another reviewer: "The mechanics part is excellent, but the theory vanishes as you reach the electricity and magnetism part." He continues: "I thought the mechanics part was pretty good, but the electomagnetic part just had bunch of equations without enough explanation as to what they mean."