# Calculus based physics differences algebra based physics

• thharrimw
In classical dynamics you will be integrating over a curved surface.In summary, algebra based physics is learning physics that pretty much only uses algebra (some trig). Calculus based physics uses calculus to derive formulas. Calculus based physics is more complicated and requires more mathematical knowledge.f

#### thharrimw

in my high school physics class we are doing algebra baced physics but i have heard of calculus based physics and i wanted to know the differences between them and the different Applications that they have.

Algebra based physics is learning physics that pretty much only uses algebra (some trig). Calculus based physics uses calculus.

So basically your class is an intro to calc based physics. You learn all the concepts of physics without messing around with all the math. If you don't know any calculus, then it will be hard to explain the benefits of it. But if you've taken calculus, then it should be pretty obvious what it can be used for.

Let me try to explain anyway, because that didn't really answer your question. Calculus let's you "add up" small contributions to get a total. For example, you have your equation for how long an object stays in the air when you throw it, right? There should be a variable there that is squared. In one of those formulas, anyway. So the only way you can do that right now is to use the formula. With calculus, you start with something more basic, and you can actually derive the formula.

Okay, that was a bad explanation. Umm... just wait until someone gives you a better one. :(

Adding calculus to Physics I, at least at the level of Halliday, Resnick, and Walker, results in hardly any new physics being taught. For instance in algebra based physics, you take the following formula as given:

$$x(t)=x_0+v_0t+\frac{1}{2}at^2$$

In calc based physics you derive that formula by integrating $\frac{d^2x}{dt^2}=a$ twice. Kinematics with nonconstant acceleration is relegated to the Exercises. What really makes calc based Physics I different from algebra based Physics I is not the calculus, but the use of the dot and cross products (most algebra based physics courses don't teach this).

Now when you get to Physics II, the calculus makes a huge difference, because you can finally learn Maxwell's equations.

I suppose for physics I, calculus makes little difference, as what is derived by calc, can often be derived alebraicly too, but calc is probably simpler. Without calculus you can't go very far in physics. Or just about anything that uses math- business, engineering, social sciences etc.

I've been told that algebra based physics is an ugly mess whereas calc based physics is nice, elegant, and actually easier. This was told to me by my precalc teacher.

Not really. Neither of them are really a mess. Algebra-based physics is actually simplified from calc physics, but calc physics has the advantage of using more elegant notation.

in algebra based phys, you let the partition of finite difference and summation goes to 0, you will get a calculus based physics. Nothing else more than that

Adding calculus to Physics I, at least at the level of Halliday, Resnick, and Walker, results in hardly any new physics being taught.
I just dusted off my very old Halliday and Resnick, copyright 1966, and they certainly went well beyond algebra-based physics. Here are a few topics in which calculus played an integral role: Work as a line integral; the rocket equation; coupled, damped, and forced harmonic oscillators; simple fluid dynamics and thermodynamics. Moreover, many topics which are presented as givens in pre-calculus physics are derived in that ancient version of Halliday and Resnick; e.g., Kepler's laws of motion. Has the Halliday, Resnick, and Walker text dumbed things down since I went to school back in the stone age?

thharrimw, This enhancement of details is how physics education progresses. You will learn some simple aspect of a problem at one level, such as the behavior of a particle subject to a constant acceleration. Calculus-based physics throws out all those seemingly unrelated formulae you learned in algebra-based physics, replacing them with a smaller set of more abstract and more mathematically advanced equations. Junior level classical dynamics throws that simple freshman-based physics out the window. Graduate level courses throw out the simple junior level stuff.

I have not yet touched on electricity, or quantum mechanics, or gravitation. The same processes occur there that occur with classical dynamics. Each step up you are learning some new physics. You are also relearning the physics you already know, but with the added twist of mathematical techniques that you presumably did not have knowledgeof the first time around.

D H,

Yes, Halliday and Resnick & Co. have watered their book down considerably since the old days. I once saw an early edition of their book that showed a derivation of the differential form of Maxwell's equations from the integral form. Now, only the integral form remains and the differential form isn't even mentioned.

Work as a line integral;

This is presented, but in the exercises students are only asked to integrate along straight line segments. And for most exercises, no integration is required at all.

the rocket equation;

This is presented, but students don't actually have to do any calculus in the exercises.

coupled, damped, and forced harmonic oscillators;

The differential equations for these systems are presented, as well as their solutions. But they don't solve the equation in the book, and the students are never asked to do it. A curious student could plug the given solutions back into the diff eq to verify that it is indeed a correct solution, but this is never asked of the student.

simple fluid dynamics and thermodynamics.

The syllabus at the school where I taught as a grad student excluded these topics, as full courses in each subject were offered. So I never went through these chapters of H&R.

ok all of this has made me more confused,
What is derivation?
What is a differential form?
and
What is a integral form?
also in calc baced physics can you use these differntial and integral things and do more then you could if you used algebra equations?

ok all of this has made me more confused,
What is derivation?
In this context, derivation is "a sequence of statements (as in logic or mathematics) showing that a result is a necessary consequence of previously accepted statements" (from http://www.merriam-webster.com/dictionary/derivation" [Broken]).
Algebra-based physics is chock full of a bunch of disparate, ad-hoc formulae that must be memorized. Many of these ad-hoc formulae can be derived from a small set of seemingly simple equations. In algebra-based physics, the expression $x=x_0 + v_0 t + 1/2at^2$ is one of those ad-hoc forumulae. It can be derived from $d^2x/dt^2 = a$, which in turn is a consequence of Newton's second law.

Another example is Kepler's laws. You probably had to memorize these as equations that just popped out of the blue in Kepler's mind. Kepler's laws are the result of deeper physics and more advanced math.

What is a differential form?
and
What is a integral form?
This is a differential form: $\nabla \cdot \mathbf{B} = 0$. The corresponding integral form is $\oint_S \mathbf{B} \cdot d\mathbf{S} = 0$. In English, there are no magnetic monopoles. If you haven't had calculus, that looks like gibberish.

also in calc baced physics can you use these differntial and integral things and do more then you could if you used algebra equations?
Yes. That is one of the reasons we use it.

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I haven't had calc but I get the concept now even though I have no idea how to do the math!Are there rules for differential and integral forms like there are in algebra? If so what are they?

if it's a toss up between a decent teacher teaching the calc based, and a crappy teacher teaching the algebra based, take the calc based.

to make a long story short, during my freshman year, I took the algebra based physics course, struggled and got a C. Then I toook the calc based physics, didn't struggle, and got a B. I was taking calc at the same time as physics, and didn't get held up by the calc...

guess that's it...best of luck with you decision.

Why do educators still teach algebra-based physics when they could simply teach the calculus mathematics first, then afterwards go straight to calculus-based physics? It saves the trouble of having to memorise equations when doing algebra-based physics.

The "rules" for differential forms come from the divergence theorem and Stokes' theorem.

$$\int_V {\nabla \cdot \vec{F}} = \int_{\partial V} {\vec{F} \cdot d\vec{S}}$$
and
$$\int_S {\nabla \times \vec{F}} = \int_{\partial S} {\vec{F} \cdot d\vec{l}}$$

Which allow you to switch from differential and integral forms. Of course, you'll have no idea what this means.

Why do educators still teach algebra-based physics when they could simply teach the calculus mathematics first, then afterwards go straight to calculus-based physics? It saves the trouble of having to memorise equations when doing algebra-based physics.

Some curricula don't require calculus-based physics, let alone requiring calculus.

Curriculum for high school or college? Calculus is an essential mathematical tool for just about every scientific, engineering field. And why don't require it when just about every student in high school is going to have to learn calculus anyway, unless they've chosen to major in the arts and literature?

Curriculum for high school or college? Calculus is an essential mathematical tool for just about every scientific, engineering field. And why don't require it when just about every student in high school is going to have to learn calculus anyway, unless they've chosen to major in the arts and literature?

Some high schools don't require physics [unfortunately]... and, if it is offered, it won't be calculus-based since calculus would be taught in the senior year, if at all. (If the high-school follows a "physics-first" curriculum, it certainly won't be calculus-based.)

Now for college...
I agree calculus is essential for science and engineering... but, as you've observed, not for a major in the arts and literature... although it does help round out a student in a liberal arts institution. In addition, I would guess that there are more non-science majors than science-majors in college. So, there is a need for an algebra-based class... although in an ideal scientifically-minded world there would only be a calculus-based one.

I was at one school that had three levels of introductory calculus-based physics... for bio and premed majors, for chem majors, and for physics and math majors. I guess that school saw the need to give the appropriate attention depending on the needs of the student, as well as the resources to devote to it. In a similar way, some schools will have algebra-based and calculus-based intended for less- and more-scientific majors.

This discussion highlights one of the main difficulties in professional training and education. Because science is constantly advancing, it takes longer and longer to gain mastery of the relevant material. Also, it leads to increasingly narrow specialization by practicioners. Most of what I have to say is for the US educational system- the European system is different, and AFAIK, students are tracked into professional/vocational programs at a very early age.

So, why not teach calculus in high school? Two main reasons- first, the teachers are not sufficiently trained in the material. Second, why teach it? Given that a tiny fraction of K-12 students go into fields requiring proficiency in calculus/physics, especially as compared to say, having proficiency in the english language (or a foreign language!).

What's the difference between calculus based Physics I and non-calculus based Physics I? Primarily conceptual. For both classes, students are expected to memorize certain formulas and are expected to plug-and-chug to solve problems. Using calculus allows for a simpler way of introducing time-dependent things (and later, spatially dependent things), at the cost of having to learn a whole new block of irrelevant math: I can't speak for anyone else, but I stopped doing "delta-epsilon" proofs and all that nonsense freshman year.

Personally, I think science curricula in K-12 needs an overhaul, and undergraduate Physics programs are also in need of an overhaul. Both are outdated products of the 60s and 70s.

Curriculum for high school or college? Calculus is an essential mathematical tool for just about every scientific, engineering field. And why don't require it when just about every student in high school is going to have to learn calculus anyway, unless they've chosen to major in the arts and literature?

My high school didn't require us to take calculus (I did anyway, apparently I was a year ahead of everybody or something). The "regular" track ended with trig.

Anyway, the physics was basic algebra-based physics.

F = ma, my mass is 5kg and my acceleration is 4 meters per second per second. What force is being applied?

Stuff like that.

Why do educators still teach algebra-based physics when they could simply teach the calculus mathematics first, then afterwards go straight to calculus-based physics? It saves the trouble of having to memorise equations when doing algebra-based physics.

I don't think there's any more equations to remember. The constant acceleration equations are used so often in introductory physics, calculus or not, that it's useful to remember it anyway. I don't think there's much more to remember.

To echo what others have said, there hardly is a difference. It makes some things easier to do, but all in all, it reduces to algebra. You might get problems with varying work, and have to integrate, or look at a graph and find the area under the line (usually the lines make triangles, so don't really need calculus) or do some derivatives to find maximum values, so not much a difference. It manly gives you different ways to do problems.

To really udnerstand physics, i think you have to understand calculus, but calculs largley came from physics so they are intertwined. Just about all physics equations are dervied with some help from calculus. It allows for more realistic problems to be solved, but as far as high school physics, you don't really need it.

It can be derived from $d^2x/dt^2 = a$, which in turn is a consequence of Newton's second law.

Say what? That's the definition of acceleration. How does it follow from Newton's second law?

I just dusted off my very old Halliday and Resnick, copyright 1966, and they certainly went well beyond algebra-based physics. Here are a few topics in which calculus played an integral role: Work as a line integral; the rocket equation; coupled, damped, and forced harmonic oscillators; simple fluid dynamics and thermodynamics. Moreover, many topics which are presented as givens in pre-calculus physics are derived in that ancient version of Halliday and Resnick; e.g., Kepler's laws of motion. Has the Halliday, Resnick, and Walker text dumbed things down since I went to school back in the stone age?

Sorry, I know this is off topic- but this sounds like a really nice book (Halliday et al.), do you have any idea where I could get the book? And, for that matter, would you recommend it?

as someone stupid enough to sign up for courses without checking to see if they'll actually go towards my degree (i know, I'm a ****ing idiot) and thus ending up taking both algebra and calculus based physics, i can tell you that there's not a hell of a lot of difference. you learn all the same concepts and equations: in my experience there was absolutely nothing new that i picked up in calc based physics. the classes only varied in that the prof spent more time going over the whys and hows of the equations and how they worked (a lot of which derived from calculus, like s=a/2(t^2)+vi(t)+si ). that's the way it is at my college at least, there could be huge differences at other schools but in my experience there was virtually no difference. IMO, you could take a class in algebra based physics, take a calculus course afterwards, and be just as well off as someone who took both calc-based physics and calculus itself at the same time (hell, you could be better off: calculus makes a hell of a lot more sense when you're learning it if you already know a thing or two about velocity and acceleration)

Calculus based physics sounds easier.

Say what? That's the definition of acceleration. How does it follow from Newton's second law?

I don't know much about cal. based physics but, I'm guessing it comes from 1/2 a^2 +m

I don't know much about cal. based physics but, I'm guessing it comes from 1/2 a^2 +m

That makes absolutely no sense. And the units don't match up either.

Say what? That's the definition of acceleration. How does it follow from Newton's second law?
I think D H is talking about this version of Newton's second law $$F=\frac{d(mv)}{dt}$$. I'm not sure how $\ddot{x}=a$ comes from this though; perhaps I'm missing something!

Basically with calculus we are able to expand upon the ideas presented with an algebraic approach to physics. Not only can the algebraic equations be derived using calculus but there are some cases where it is much more practical (and easier) to use calculus.

For instance say we wanted to find a velocity of a function at a certain time, with only knowing it's position at any given time. Without calculus the best we can do is approximate this. But since a velocity is just a change in position, if we find the change in position over an infinitely small time interval we can find the actual velocity of an object. This would be an example of differentiation.

An example of integral calculus would be something like this. Say you have a rigid rod and you wanted to calculate the force of gravity the rod exerts on another object at sometime. Well to do this we need to chop the rod up into finitely small parts and find the force for all of these parts, then sum them together to get the total force. Without calculus goodluck summing up the force of an infinite number of pieces of a rod.

Also you can have differential equations (Just shows how a particular function is changing) and you might want to calculate a value of the function at a particular point. A good example of this would be a spring that is dampened.

algebra based physics is what pre med students take
calc based physics is what science/engineering majors take.
simple as that.

That makes absolutely no sense. And the units don't match up either.

Oh the plus should be a x and the m should be a v. I just did some basic antidifferentiation.