14
I took an excellent set of related rates problems from Jim Belk's webpage and turned it into a nice worksheet. I spun the problems this way, to motivate them:
We have now covered related rates. This means you now have an extra power that works in all your science classes. Anytime anyone gives you an equation -- a fact -- that is true about the world, you ...
13
Better don't learn such conversion factors, learn how to derive such on the fly. This helps also in the case you want to change inches per second into feet per hour, or cubic feet per hour into cubic centimeters per second.
12
Here's one that I just thought of, by modifying a problem in the ODEs section of my textbook.
Question: In the presence of air resistance, does a thrown ball take longer to go up or to come down?
We assume that the force of air resistance is proportional to velocity, with constant of proportionality $p$. Thus, we have
$$ F = m a = - p v - m g. $$
Since $...
11
First f all, check out this great paper. It has some interesting examples, especially in Appendix A.
A typical example is relativistic mechanics: the usual first order approximation yields the unsatisfactory Newtonian mechanics (see for example here), which is in many applications not enough to explain what is happening.
ADDED: Since OP asked for an ...
10
Much of quantum mechanics is only known in a perturbative framework. Essentially, the basic objects of interest are series. Terms are calculated by Feynman diagrams of increasingly complex diagrams for higher order corrections. So, the series description is pragmatically fundamental. I suppose in principle, non-pertubative solutions exist, however, if no ...
10
This is not exactly what you seek, but at least you can find educational research articles
under the key phrase dimensional analysis. Below [1] says it's useful, [2] questions that usefulness.
[1] Hrayr Ohanyan. "The Application of the Method of Dimensional Analysis When Solving Problems."
American Journal of Educational Research.
Vol. 4, No. 1, 2016,...
9
There are no clearly defined mathematical prerequisites that are needed in order to learn quantum mechanics. QM can be taught at a variety of levels. Here are some examples of levels at which it can be taught:
There is a very nice book by Hewitt, Conceptual Physics, that
presents a complete survey of physics for a gen ed course with no
more mathematics than ...
8
As celeriko took the what I consider as what "ambient" space the object lies in approach (I.e., Whittney's embedding), I'll consider the degrees of freedom of movement.
Consider the normal directions you would have someone move when they are standing straight up. Have the child walk back and forth in a straight line. We can consider this the forward-...
7
To answer the question
Is there any real life situation where you can intuitively see that the km/h number is higher?
and seeing that you are based in Germany:
The ICE goes up to 300 km/h, but nobody suggests that it comes anywhere near the speed of sound.
This, of course, presupposes that one knows the rough speed of high speed trains in km/h, and ...
7
Unfortunately, dimensionality is a very tough concept to truly understand, even for adults. I know that you are trying to find a clear and simple method, which what I have typed below is certainly not. My hope is you can use the information below to get ideas on how to develop your own scaffolded method for your student. What I would suggest would be a ...
7
It might help to make vivid what this conversion means. 10 km/h is a reasonable running speed. A soccer field is 105 m long. In one second, can you run a few meters, or can you run a third of a soccer field? (American football fans might have a better sense for this, as the number of yards traveled by a runner is key to the game, but I assume that soccer is ...
7
I don't know if this is the sort of thing you're thinking about -- it seems to me like a too-obvious example, but it seems to fit your question pretty well. Suppose you have a particle moving along some path $r(t)$, and suppose we expand $r(t)$ in a Taylor series as $r(t) = a_0 + a_1 t + (1/2) a_2 t^2 + (1/6) a_3 t^3 + ....$ Then $a_0$ tells us the initial ...
7
Here is a list of how I actually used linear algebra as a physics major. The general techniques are
Change of basis (including calculating reciprocal vectors, which in turn require calculating determinants)
Projections onto subspaces
Finding the eigenvalues and eigenvectors (including calculating determinants)
Some examples
calculating the intersections ...
6
I have two good examples, they are somewhat similar, but one is solved more analytically, and the other more computationally:
Masses on a spring:
From classical mechanics: a mass on a spring with one end fixed can be described by the classical equation:
$ \mathbf{F} = m\ddot{\mathbf{x}} = -k\mathbf{x} $
If we have two masses on a spring, (in 1d)
$ m_1 \...
6
Flatland. W. Abbott, 1884
Here's the Project Gutenberg link to read or download in your preferred e-version:
https://www.gutenberg.org/ebooks/97
As an advanced student in elem school, my engineer single-parent mother gave me a copy while I was surviving a two year nothing-to-learn transition into public school from lifelong private. I believe I was 11 when ...
6
While I did not attend any elementary physics courses by mathematicians for mathematicians, I did attend a curriculum of courses called "Mathematical Physics" held by mathematicians for mathematicians and physicists. In terms of physics and mathematics, they were some kind of advanced level course - it helped if you knew linear algebra, calculus and the ...
5
As stated above QM can be taught and understood at a variety of mathematical levels.
At a minimum the required concepts are:
1) Linear Algebra
2) Probability Theory
3) Calculus (basic derivatives and integrals, multivariate would be even better)
4) Newtonian physics and vector mathematics
Now all of these topics can be explored at a variety of levels ...
5
As for almost any course, the best person to ask is whoever is teaching the following course. They are the only ones who really are in position to know what is needed.
Yes, it will happen that they forget to mention some critical topic. And very, very often what they tell you is that "they should know about ..." when they really require the students to be ...
5
In the paper Research on teaching and learning Mathematics at the Tertiary level, by Biza, Victor-Giraldo, Hochmuth, Khakbaz, Rasmussen, recently published by Springer (ICME 13 Topical Surveys) you find at least some evidence supporting your professors' attitude. It is said there, in fact, that analyzing the way students use mathbooks, and in particular ...
5
I am going to assume that you are teaching a calculus "helper" versus the entire physics class. Your initial statements don't match that. But then all your content described is math, not physics. And also 50 minutes per week sounds rather light for a whole class. [If the converse is the case, I would spend your time on...physics.]
With that in mind, my ...
4
All the above good answers aside, I think that this question misses the mark in so far as the real problem is to internalize a good notion of what 'dimension' means. In particular, someone that age who familiar with the science-fiction notion of 'dimensions' referring to parallel universes (eg, traveling between dimensions with a magical vortex or portal) ...
4
You'll want to keep it simple and visual. Use easy, real-world examples.
I would say a dimension is similar to a "direction you can move in".
On a string you can only move left and right. It has one dimension. On a piece of paper, you can move left and right, but you can also move up and down--two dimensions. In a fish tank you can move up, down, left, ...
4
How many suvat equations do you use? I have taught, in two countries in multiple schools using multiple books, four equations of constant acceleration. That seems to be standard. Each equation leaves out one variable, and the four standard equations have one question each leaving out $s$ (position), $v$ (final velocity), $a$ (acceleration), and $t$ (time ...
4
What I try to emphasize is that any formula they're familiar with requires integral calculus once some of the quantities vary.
Basic example: (distance)=(rate)*(time). How far do you go if your rate is a function of time?
Medium example: (force)=(mass)*(acceleration). What is the end velocity of a rocket that burns mass (i.e. fuel) to exert constant force?
...
4
In special relativity, we have $\gamma=(1-v^2)^{-1/2}$, where $v$ is the velocity in units of the speed of light. Relativistic momentum for a particle with $m\ne0$ is $p=m\gamma v$, and kinetic energy is $K=m(\gamma-1)$ (in units where $c=1$). (a) Expand $p(v)$ in a Taylor series and show that the lowest-order nonvanishing term recovers the nonrelativistic ...
4
While this isn't a 'mainstream' topic, I feel like homotopy analysis methods are worth a glance here. The idea is to approximate some non-linear equation by a linear one, take a continuous connection between them, solve the linear problem, and 'slide' the solution over to the non-linear case. If $L\{u(t)\}=0$ is the linear operator and $N\{u(t)\}$ is the non-...
3
I have faced this teaching problem before, but I am not very experienced a teacher.
This is what I think right now.
Rather than try to get them to memorize both the factor (3,6) and the "side" (multiply from m/s to km/h) it might be better to have a standard conversion that they think about each time (till it becomes automatic)
"If I walk 1 meter each ...
3
This should rivet students' attention!
See the MathOverflow question,
"The “Dzhanibekov effect”--an exercise in mechanics or fiction? Explain mathematically a video from a space station,"
and especially the beautifully informative response by Terry Tao here.
It is also known as the Tennis racket theorem (Wikipedia).
(The remarkable) video URL here:
...
3
You might be interested in Brian C. Hall's book Quantum Theory for Mathematicians. The author writes on his webpage about the book:
This book aspires to be a self-contained and reasonably comprehensive treatment of quantum mechanics (excluding quantum field theory) from a mathematical perspective. No prior knowledge of physics is required, but only the ...
2
In classical mechanics, the motion of a rigid body can be described completely by the motion of the center of mass and by the rotations around the center of mass.
As opposed to linear velocity and momentum, the angular velocity does not always point in the same direction as the angular momentum. As mentioned on this site, this can lead to problems as it ...
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