How is Physics taught without Calculus?

[renormalize]

Isn’t that still using the “product rule” (calculus) in a way?

Well, an approximate "product rule" doesn't really require calculus. It falls out directly for small finite-differences:$$\Delta x^2 \equiv (x+\Delta x)^2-x^2=2\Delta x+(\Delta x)^2 \approx 2\Delta x\text{ (for }\Delta x\text{ small)}$$

 

[jack action]

You should use the average force $\bar{F}$:
$$W = \bar{F} \Delta x = \left(\frac{k \Delta x + k (0)}{2} \right)\Delta x = \frac{1}{2}k (\Delta x)^2$$

 

[jack action]

How do you convince someone of the validity of the second equation without alluding to the “product rule”?

Is it really that long to show this extra step explaining the difference in area:
$$\Delta (PV) = (P+\Delta P) (V+\Delta V) - PV = P \Delta V + V\Delta P + \cancelto{0}{\Delta P \Delta V}$$
With a drawing like this:

https://en.wikipedia.org/wiki/Product_rule#Discovery

 

[PhDeezNutz]

It is not….honestly that makes sense. I don’t recall ever seeing a diagram like that.

 

[gmax137]

I don’t recall ever seeing a diagram like that.

Diagrams like that are worth their weight in gold. I'm pretty sure that's how my high school calculus math teacher "explained" the product rule.

 

[PhDeezNutz]

Diagrams like that are worth their weight in gold. I'm pretty sure that's how my high school calculus math teacher "explained" the product rule.

I’m pretty sure it was explained to me in that manner at some point and I just forgot. I can’t conceive of my past teachers (all of whom were great imo) would have just glossed over the intuitive explanation for the power rule presented 2-3 posts above.

You are absolutely right though: an explanation like that is worth it’s weight in gold.

 

[hutchphd]

I will now kick the hornet's nest:

more plausible approximate equation for the work done by the spring restoring itself is:

I don’t recall ever seeing a diagram like that.

It was taught in your calculus class. To do physics one must "use calculus". To teach physics well you will either need to either call it out as a prereq or teach it concurrently Whether you utter the offending word "calculus" or not, somehow the material must be conveyed.

Nest kicked....



.

 

[PhDeezNutz]

I will now kick the hornet's nest:





It was taught in your calculus class. To do physics one must "use calculus". To teach physics well you will either need to either call it out as a prereq or teach it concurrently Whether you utter the offending word "calculus" or not, somehow the material must be conveyed.

Nest kicked....



.

I presume by saying "nest kicked" you meant to stir the pot. However I don't take issue with anything you said. In fact, I wholeheartedly agree with it. Moreover "finding how to teach things without calculus" has made me reflect on my own understanding.....which is important if you want to convey it to others.

 

[hutchphd]

Absolutely. The only physics I truly understand are those parts that I was fortunate enough to teach. This is because I am really lazy.
The hornet's nest comment was engendered by the previous lengthy and somrtimes supercilious) discussion which seems not to have reignited!

 

[Muu9]

I will now kick the hornet's nest:





It was taught in your calculus class. To do physics one must "use calculus". To teach physics well you will either need to either call it out as a prereq or teach it concurrently Whether you utter the offending word "calculus" or not, somehow the material must be conveyed.

Nest kicked....



.

I don't see how a diagram like that couldn't be included in a non-calculus based physics course, even if only as a sidenote ("if you want to know why this formula is true, you'll need calculus, but here's the gist of it. No, this won't be on the test.")
Buzz Buzz

 

[gmax137]

I think the "calculus-y" part is when you say "we can ignore $\Delta u \Delta v$" -- but you can believe even that from the diagram.

 

[hutchphd]

No, this won't be on the test.

What does that mean? You need only memorize this formula but not understand it? Is it any wonder the hapless student is forever seaching for "just the right formula"?? One of my least favorite pedagogical questions "will that be on the test?"

 

[Muu9]

You need only memorize this formula but not understand it?

Yes; it's unrealistic to expect the average student in the algebra-based physics class with no calculus experience to understand that diagram as anything other than a somewhat intuitive mnemonic

 

[Will Flannery]

When I was a freshman at UF taking an honors physics class I thought, gee, why are we learning Newtonian physics when Einsteinian physics is more than half a century old? When I graduated with a PhD many years later and went to work analyzing rocket trajectories for the Star Wars program, I learned that I didn't learn Newtonian physics in the University, I learned what we can call rudimentary physics.

Since the time of Newton classical physics has consisted of the analysis of differential equation models of physical systems. Since differential equations are an advanced topic in calculus, physics is necessarily taught at the U for two years ! without differential equations.

Unfortunately, once you've taken a course in differential equations, the real difficulty with Newtonian physics presents itself, that is, most differential equations are unsolvable.

Thus Kepler's Problem, which Newton (and Kepler) solved (numerically using Kepler's equation) is not taught in University. To see how difficult the analytic solution is, check wiki Freefall to see the analytic solution in 1-D, an infinite series solution.

From Thornton's Classical Physics - "Edmond Halley is generally given the credit for bringing Newton's work of gravitational and central forces to the attention of the world. After observing the comet personally in 1682 Halley became interested. Partly because of a bet between Christopher Wren and Robert Hooke. Halley asked Newton in 1684 what paths the planets must follow if the Sun pulled them with a force inversely proportional to their distances. To the astonishment of Halley, Newton replied "Why, in ellipses, of course." Newton had worked it out 20 years previously, but had not published the result. With painstaking effort Halley was able in 1705 to predict the next occurrence of the comet now bearing his name, to be 1758."

As I learned working on Star Wars, orbit problems are easy to solve numerically (from the laws of motion and gravity), i.e. simulate, and that's the way it's done in engineering today. For details see ... The Coming Revolution in Physics Education ... https://www.academia.edu/42129766/The_Coming_Revolution_in_Physics_Education

 

[Haborix]

Physics without calculus is like eating an appetizer. Very nice, but should be getting you excited for the next course. But you won't be worse off if you never make it to the main course.

Regarding simulation: time should not be wasted in core physics classes on simulations. There is enough material to get through without spending time teaching people to simulate things (a task which I think is best left to specialized course, the computer science department, or as personal learning on the side).

 

[Will Flannery]

"Regarding simulation: time should not be wasted in core physics classes on simulations."

If we step back a minute, what is physics? Ans: creating mathematical models of physical systems and analyzing the model to be able to predict how the physical system performs.

In classical physics, the models are sets of differential equations. and ideally the model is analyzed by solving the differential equations. Given the solution, we can predict how the real system will perform.

However, the differential equation models of most physical systems are not analytically solvable. Thus, we read in 'Deep Learning for Teaching Physics to Computers' (satirical but accurate) by former AJP Editor R. Price "At Crenshaw-Mellon University,9 in fact, simple computer programs have been developed to recognize and solve the dry-sliding-friction-block-on-tilted-plane, ballistics, and pendulum problems that constitute almost all of university physics."

What are the goals of simulation? The goal of a simulation of a physical system is predicting how the real system performs. What does a simulation start with? It starts with a differential equation models of the system. So, the goals of the classical analysis of a physical system and the simulation of the system are the same. The difference is that simulation can be used to analyze analytically unsolvable systems. That is why computational calculus, i.e. simulation, has been the norm for the analysis of physical systems since the mid-20th century.

 

[Herman Trivilino]

Up until about 1960 or so the introductory college-level sequence for physics majors was taught without the use of calculus in America. To the uninitiated using calculus to introduce physics is rather like using magic. Students, on average, show little comprehension of even the most basic concepts and instead rely on memorization to pass tests. Anyone who has ever taught introductory physics, with or without calculus, will recognize this affliction.

Edit: If you look at the textbooks used prior to about 1960 to teach introductory college-level physics to physics majors you will see that they rather resemble the noncalculus introductory college-level physics textbooks used today.

 

[gleem]

Up until about 1960 or so the introductory college-level sequence for physics majors was taught without the use of calculus in America.

Huh! The text I used in 1960 was by Shortey and Wiliams published in 1955 meant to be used with a concurrent course in calculus. I know this text was used by Wheeler at Princeton. Since the college that I attended was of no particular national importance I find it hard to believe that physics was routinely taught without calculus to physics majors during this time.

From the preface of the Australian edition of Sears and Zemansky University Physics

When the first edition of University Physics by Francis W.Sears and Mark W. Zemansky was published in 1949, it was revolutionary amongcalculus-based physics textbooks in its emphasis on the fundamental principlesof physics and how to apply them

 

[Mordred]

You would be surprised. For example the high school I attended didn't even offer Calculus. I ended up taking it as an option when I went to University. Glad I did as I needed it for Particle Physics.
Though not as much for Cosmology or Astrophysics. (Never took Astrophysics for the record). That was a good 35 to 40 years ago though lol.

Now I meet students that graduate that cannot divide fractions

 

[Herman Trivilino]

Huh! The text I used in 1960 was by Shortey and Wiliams published in 1955 meant to be used with a concurrent course in calculus.

I remember reading this in a journal years ago but must have got the year wrong. Perhaps by several decades.

I used the same text in 1973-74. The professor didn't use calculus even though it was a co-req. I also took calculus and a calc-based physics class my senior year in hs.

 

[halalaal_hi]

I would say that physics cannot be truly understood without calculus. I took non-calculus physics and calculus I at the same time. My physics book basically just told us formulas and equations whose derivations were "beyond the scope of this course". These were such things as gravitational potential energy, springs, centripetal acceleration, etc which can only be explained with calculus. With just calculus I, I was able to derive these and some things that weren't even mentioned in the book.

 

[weirdoguy]

which can only be explained with calculus.

Derived. They can be explained without calculus. And I can assure you - a lot of high school students do not grasp the physical meaning of the formula just by looking at the derivation.

My physics book basically just told us formulas and equations

Than it was a bad book.

My polish non-calculus high school students understand physics way better then my english calculus-based high school students, because in the latter case books focus way too much on using derivatives in physical context than on the physics itself.

I would say that physics cannot be truly understood without calculus.

Physics in general - yes. I don't know, maybe I misunderstood the premise of this thread. At some level it can be understood, as years of my (and not only my) teaching shows. Besides, physics olympiad (again, polish one) shows directly that knowing derivatives does not equal understanding physics. 99% of tasks is calculus free and is very hard, even for me. I still learn to be able to prepare people for that. And I know that classical fields are sections of jet bundles But that does not help

 

[Herman Trivilino]

I would say that physics cannot be truly understood without calculus.

True, because calculus was in large part invented to develop physical laws. But that process was completed by seasoned researchers not by beginning gen ed students.

I took non-calculus physics and calculus I at the same time. My physics book basically just told us formulas and equations whose derivations were "beyond the scope of this course".

The vast majority of college-level non-calculus books derived all those equations. Of course the derivations that use calculus are not convoluted, but students must understand calculus to appreciate that. And even calculus students are struggling too much to be able to appreciate.

These were such things as gravitational potential energy, springs, centripetal acceleration, etc which can only be explained with calculus.

Not true. See just about any college-level non-calculus books true. See just about any college-level non-calculus textbook.

 

[Astronuc]

Up until about 1960 or so the introductory college-level sequence for physics majors was taught without the use of calculus in America. To the uninitiated using calculus to introduce physics is rather like using magic.

I think that the situation depends on the university, the faculty (in the physics department) and the quality of students (i.e., students' preparation). Certainly, when I studied introductory physics at university, and even high school to some extent, the texts included calculus. However, I was aware of physics courses that at most used algebra and geometry.

Some Landau and Lifshitz texts were available in the 1950s.
https://en.wikipedia.org/wiki/Course_of_Theoretical_Physics

I think it fair to say that by the 1960s, there was an increased emphasis in using calculus in introductory physics programs.

 

[Herman Trivilino]

I think that the situation depends on the university, the faculty (in the physics department) and the quality of students (i.e., students' preparation). Certainly, when I studied introductory physics at university, and even high school to some extent, the texts included calculus. However, I was aware of physics courses that at most used algebra and geometry.

Yes, as I explained in Post #110 I got my dates wrong.

I think it fair to say that by the 1960s, there was an increased emphasis in using calculus in introductory physics programs.

That's true. Feynman discusses this in the introduction to his famous "Feynman Lectures in Physics" written sometime in the early 1960's IIRC.

 

[Vanadium 50]

And before Newton, physics was never taught with calculus!

We've been teaching physics with calculus for over 60 years. There must be a reason, other than "to torment students". It's because a) it is easier - fewer equations need to be memorized or appear out of thin air, and b) it allows for the student to solve a wider range of problems.

 

[ohwilleke]

The question is not so much "can physics be taught without calculus" as what reasonable goals are for a teacher teaching physics without calculus.

Without calculus you want to focus on conceptual and qualitative ideas, rather than doing calculations.

For example, you can explain that the movement of perceptible physical objects (as opposed to subatomic particles) can be determined more or less exactly by considering all forces acting on it, and you can explain different kinds of forces, and explain how friction muddies the waters.

You can bring in the equation of Newtonian gravity, so that students understand what factors influence it (the respective masses and their distance from each other) without calculating exactly how much motion is caused by gravity or mathematically proving Kepler's laws from that formula as you might in a calculus based course.

You can explain what induction is in electromagnetism without actually pulling out Maxwell's equations.

You can build intuition about how things like leverage work, by doing hands on experiments that will be remembered, rather than doing calculations on paper that mathematically weak students will forget.

You can explain some of the basic ideas of thermodynamics with drawings and hands on experiments, rather than equations.

Rudimentary physics concepts like these are usually taught to some degree starting in elementary school.

Non-calculus based physics isn't sufficient to be an engineer, but it is useful in developing a scientific worldview that can be refined later.

 

[hotvette]

Wow, does this bring back a memory. My high school had no calculus (in 1972), so physics was non-calculus based. When I got to college I put off taking calculus until second semester, but took a calculus based physics class (by mistake) my first semester. I was totally lost. I went back to my high school physics teacher for help and he told me to just think of $\frac{dx}{dt}$ as just $\frac{\Delta x}{\Delta t}$ like we learned in high school physics. It was enough to get me through the class. I started my calculus sequence the next semester.