1) You are driving a car, and you accelerate from 0 to 25 mph. Say the amount of energy required is x. Now you continue to accelerate from 25 mph to 50 mph. What is the total amount of energy you used to accelerate from 0 to 50 mph?

OK, my intuition would be 2x. Because if you're accelerating at the same rate, it would take x energy to go 0-25 and then x more to go from 25-50. But that's wrong. It's actually 4x. That seems really unintuitive, which prompted me to do much thinking to reconcile that.

The weird thing (to me) is that an object moving twice as fast as another object (of the same weight) has four times the energy, not two. The way I finally got an intuitive understanding of that is to think of dropping an object from ten feet, and it having a velocity of y when it hits the ground. Now to get the velocity to be double that, you would have to drop the object from four times as high, not two times. It seems intuitive that carrying an object up to 40 feet would take four times the energy than carrying it up to 10 feet. Then when you drop it from forty feet it would only have twice the velocity as from 10 feet. But it would still have four times the energy.

Damn, this is difficult to explain - have I just confused everyone so far?

(to be cont.)

Ack! Jesse, you just reminded me that I didn't bring a lunch today. (I ate all my deli turkey [intended for a sandwich] last night.)

Hmm. Am tempted to go next door to Chipotle and get a burrito bigger than my head....

Cats haven't reached terminal velocity in falls that short though, they just have more time to flip themselves over into landing position. I think they'd hit terminal velocity about the same time as similarly dense human beings... somewhere around 60 stories, isn't it?

I'm not willing to get up at the buttcrack of dawn for an office. I have skewed priorities.

Yeah, additional sleep, even just fifteen minutes or so, will make me happy. Or maybe more, depending on how I shift my hours.

Of course, once I've spent some time without an office, I'll probably be whining about how I'd trade a little sleep for my office back.

Why am I thinking about this? I believe that to really get an understanding of physics (or math, or science in general) it's much better to get an intuitive grasp of what's going on rather than just memorizing formulas. But it's weird to me that the physics of motion and energy seems so damn unintuitive, which made me obsessed with actually getting that intuitive understanding.

ita-level specificity

I thought it was totally commonly supposed to be true.

a cat thrown from a 10-story window is just as likely to survive as a cat thrown from a 5-story window, or whatever the minimum cat-flip-over-land-on-feet height is

I guess you're not of the school of thought that has the fall (and not the attempted landing) as being the first fatal bit, huh? Is that supposed to be true, or debunked? Also, how do you test and remain even vaguely ethical?

I'll probably be whining about how I'd trade a little sleep for my office back.

I wouldn't trade sleep for an office even if it meant getting away from Tiny the Loud Talking Sales Guy. Unless my iPod died.

In other news, Bill Clinton will be at the Michigan Ave. Borders in about an hour.

Um... my cat fell out of a second story window seemingly unhurt, however 1 week later he tried to kill me. I am not sure if proves or disproves the theory.

I still haven't wrapped my brain about the whole "So you're driving a car at 30MPH and your headlights rake over Albert Einstein. How fast is the light going when it hits him?" question. So, yeah, counter-intuitive is the name of the game, for me.

(Actually my problem is relative motion. Dropping a basketball from a moving car, e.g., completely blew my mind with its trajectory, even after I'd viewed the whole thing as filmed with a static camera.)

somewhere around 60 stories, isn't it?

That high? That seems -- a bigger number than I would have guessed. Also, I would have guessed it was not density but, like, air-profile that determined terminal velocity. Whatever you call the difference between an ordinary squirrel and a flying squirrel with wing-y flaps extended.

OK, a quick explanation of the car thing. A car that's moving at 25 mph actually is at a disadvantage when it comes to acceleration compared to a car at rest. Assuming a one-speed transmission, the engine is already turning pretty fast, meaning it's completing more cycles and burning more fuel. So the engine has to go faster and faster, completing more and more cycles, burning more and more fuel in the same space of time as the car approaches 50 mph. Hence the consuming four times the energy to get to 50 mph.

Has what I've said made sense? Does the whole thing just weird you out with the unintuitiveness? Or is it just seem too abstract and "mathy" to get that feeling?

I'm wondering what Nilly and Gud would think....

The subject combines two areas of fascination to me - math/physics, and the human brain's intuitive understanding of how the world/reality works. 'Cuz the two are often at odds.