Motion in Two or Three Dimensions (Physics)

I actually finished reading chapter 3, "Motion in Two or Three Dimensions," of Young and Freedman's University Physics quite a long time ago.  I try to read two pages of either this physics or a calculus text a day.  So since I read this chapter I have read the first chapter of Stewart's sixth edition calculus text and am almost done with chapter 4 of Young and Freedman.  So I want to catch up with the old before I finish chapter 4.

One observation I've made is that the broad strokes of physics and chemistry are really not too difficult to explain.  The Devil is in the details of application.  As always, I'm tempted to write something to boil things down without the burden to be technically correct.

This chapter's review list:

• The main difference between motion in 2 and 3 dimensions from one dimension is simply that one must now use vectors.  This is just same old same old, dividing out the x, y, and perhaps z components of displacement, velocity, or acceleration, whether as an average or as an instantaneous derivative.
• Projectile motion is 2 dimensional motion.  Using trig, the initial velocity x component in projectile motion will be v_0x=v₀cosθ while the initial y velocity component will be v_0y=v₀sinθ.
• The x velocity component is constant because of Newton's first law, Galileo's law of inertia, so the distance, x=(v₀cosθ)t very straightforwardly (distance=velocity x time).
• The y velocity component must deal with the constant negative acceleration of gravity, -g, which yields the slightly more involved formula for y=(v₀sinθ)t-½gt², based on the formula, d=vt + ½at²
• Motion in a circle is also 2 dimensional.  Although I don't think converting from radians to degrees and visa versa is a big deal, the reasons for radians hasn't quite clicked with me.  In circular motion with a constant speed, the acceleration always points inward toward the center of the circle.  This "centripetal acceleration" is a=v²/R
•  Since speed is distance/time, and in uniform circular motion, the distance around is the circumference, which equals πD (diameter, which is also 2R, radius) = 2πR, the velocity thus equals 2πR/T.
• Accordingly, if we substitute 2πR/T for v in the equation above, we get a=4π²R/T²
• In non-uniform motion, we have to take the derivative of the velocity to get the component of acceleration that is tangent to the circle, and the vector sum of the total acceleration will not point directly toward the center.  It will point either ahead or behind, depending on whether we have acceleration or deceleration.
• The final section of the chapter had to do with velocity relative to different reference frames.  So the velocity of a point in frame B in relation to frame A is a vector sum--the velocity of point P in relation to B "plus" the velocity of frame B in relation to frame A.

Motion in One Dimension

I think it was about a week ago or so that I declared chapter two of University Physics, by Young and Freeman, dead.  I had been sauntering through it for months.  The main points are easy, although as usual the application of them into concrete situations is more diffiicult.  But here I summarize the main points:

1. Average velocity Δx/Δt or (x2-x1)/(t2-t1).

2. Instantaneous velocity is lim Δt→0 of Δx/Δt, which = dx/dt.

3. Average acceleration Δv/Δt or (v2-v1)/(t2-t1).

4. Instantaneous acceleration is lim Δt→0 of Δv/Δt, which = dv/dt.

Then these equations work for constant acceleration
5. v₂ = v₁ + at

6. x₂ = x₁ + vt + ½at²

7. v₂² = v₁² + 2a(x₂-x₁)

8. x₂ - x₁ = [(v₂ + v₁)/2]t

9. Finally, for varying acceleration, one will have to integrate:

x₂ = x₁ + ∫0→t of v dt

v₂ = v₁ +∫0→t of a dt

Quantum Physics 1

When I was about twelve, I used to play a game every once and a while to keep me from getting bored--and I have always been very easily bored. I would pretend as if I suddenly developed complete amnesia and had to figure out where I was. Another question I used to ask myself is how much of human knowledge I could recreate if everyone else on the planet suddenly disappeared or if I were stuck on a deserted island. Not much, I always concluded.

So what if I were the last person alive on earth? I'm sure my first order of business would be figuring out food, crops, etc. I'd want to figure out medical type things and be ready for any kind of sickness. Assuming that nothing electrical worked, maybe I would spend some time trying to fix some things along those lines. Sure, I might travel too. I could live in a different ghost city every week if I wanted to.

But eventually, if I had access to all the books humanity had left behind, I would probably start to study things. Let's say one year or two I decide that I want finally to spend the time to learn quantum physics. I'd always wanted to, but didn't have the time. Of course I'd have to review a bunch of physics I hadn't thought about since high school and college, and that was now decades ago.

Where would I start?

Quantum Physics 2: Planck 1

All the introductions to quantum mechanics very quickly mention Max Planck. What do I remember of Planck from high school, college, somewhere? Didn't he suggest that maybe on the atomic level energy jumps from one intensity to another without going through all the levels in between? And I remember a formula, E=hv, where h is Planck's constant. Well, that doesn't take me very far. Better look in the library.

OK, here's a little book by John Polkinghorne published by Oxford in 2002: Quantum Theory: A Very Short Introduction. He starts a little further back than others. In the 1800's, people are thinking that light is a wave, like the sea. Isaac Newton back in the 1600s had thought light might be particles. He mentions James Clerk Maxwell as setting down the basic equations of electromagnetic theory, puts him in the same category as Newton, considers his equations the greatest discoveries of 1800s physics. They point to light being a wave.

Do I get distracted here? Do I now look for a college physics book to relearn the basics of electromagnetic theory? No one else is alive. I've got a lot of time on my hands. I'm probably going to have to go back to learn it at some point in this quest anyway. May as well.

Thursday was Physics Day

I was well pleased with myself yesterday. I have a long project going to understand one equation, Schroedinger's equation. It is an equation related to potential. So I went to a chapter on electric potential, which lead me to review in a chapter on work (W=Fs or more specifically F dot s). I ended up reviewing in the first chapter on vector notation.

The thing that made me happy was when I saw that I was looking at something called a dot product, the words "as opposed to a cross product" came to mind. I smiled, turned the page, and there it was, the cross product.

I don't remember learning these two the first time. I wonder if I understand them better this time than I did twenty-five years ago. But I was happy that somewhere in the dark recesses of my mind, I remembered something!