WOOH! I’ve decided I’ve gone through enough math to get back on the physics train. Here on out, I’m going to assume you can use derivatives.
Section 2.3: Average and Instantaneous Acceleration
If you’ll recall, long ago when we did section 2.2, it was about average and instantaneous velocity. We showed that velocity was just position changing over time. Now, we see acceleration is just velocity changing over time.
Makes sense, right? You have a position equation. You say “how’s it change with time?” You take the derivative with respect to time, and you get velocity. Now you wanna know how velocity changes with time, so you just take another derivative with respect to time. Now you have acceleration.
And, you can remember the order of that by how units stack up. Position is in meters (with respect to some point), velocity is in meters per second, and acceleration is in meters per second per second. You can also say “meters per second squared,” but I don’t really like that. “Velocity per second” makes sense to me, but I have no clue what a squared second does. I tend to say “per second per second” verbally, but use the power notation when writing it, since it’s simpler.
If you want, you can actually take the third derivative of position with respect to time, and get something called “jerk.” That’s the acceleration per second, whose units are meters per second per second per second. Jerk is also a tasty sauce, a good movie, and a fun activity.
In fact, if you’re an ultra-nerd, you can jounce. That’s the fourth derivative of position with respect to time. Its units are meters per second per second per second per second. Hence, if you go to a girl who likes physics and ask if she wants to jounce it with you tonight, she’s liable to say yes. Or, maybe she’ll call you a jerk.
And that’s pretty much all there is to this section. Let me give you two quick thoughts though:
Stephen Pinker once wrote about how humans have a sort of intuitive physics, which is kind of right, but wrong in a lot of specifics. That is, your intuitive physics tells you that if you throw a ball through space, it should slow down. Newton tells you you’re wrong and punches you in the face. Then Einstein unzips and pees on you. That’s what intuition gets you.
Similarly, I think we’re not very good at telling the difference between velocity and acceleration. They feel sort of the same, even though they have noticeably different effects. For example, say I’m running. When I slow down, which way am I velocitating and which way am I accelerating? Well, clearly I’m velocitating forward, since I’m going that way. But, I’m accelerating backward. This makes sense when you write it all out, but it’s a little weird on the brain at first, especially if you’ve never studied physics. For example, get a friend in your car, get to a high speed, then tell them “I’m going to accelerate this way” and point to the back of the car. Your friend will probably freak out, thinking that you’re going to start moving backward. But, of course, whenever you tap the brakes, you’re accelerating backward.
Getting your head on straight about these things is part of why practice is very important. The book has some pretty graphs, but one thing that’ll really help you nail down this stuff is this: Stand up, run forward, slow down. Then walk backward, and slow down. Then, see if you can quickly explain to your invisible boy/girlfriend when you were accelerating and in what direction.
The other thing I want to talk about is how, like a dick, I keep using the phrase “with respect to” whenever I talk about derivatives. This is important: “Taking the derivative” isn’t just some trick you do to equations. You’re actually asking a question of your equation. You’re saying “How do you change with respect to this other thing.” Always remember that. The derivative of your position with respect to time is your velocity. The derivative of your position with respect to temperature is not. Both of those are valid moves. In the case of mechanics, the first one is simply the more interesting one.
Next, we learn a couple awesome equations.