Whee! Sorry for the lack of updates, but I was fighting the good fight at San Diego Comicon. Christina snapped some great photos.

Since this section is fairly straightforward (though there are some important things to note), I want to take an opportunity to talk about the beauty of math and physics.

I do pretty well for myself, at least as far as geeks are concerned. I have a nice small business, I have far more readers than I deserve, and I get to talk to awesome people like Google and the kids at UCDavis. I don’t mean to complain, but there are times when it can feel overwhelming. This is particularly true at comic conventions. Even though it’s often fun, it’s exhausting to have to be “on” 24 hours a day for a week. You meet so many people, and everyone seems to have figured out how to make the next big thing or to have a connection to some important person you need to know. Although some of it is for real (neat stuff can happen at a big con), you eventually get tired of all the pageantry and bullshit.

And that’s where I love having some background in math and physics. It’s a mental place to go home to. Why? Because math doesn’t care how cool you are or what your reputation is or who you work for or any of that. Nothing superficial about you will give you any insight into a mathematical theorem. No wealthy acquaintance will give you any insight into the strangeness of the Universe. So, for me, it’s always very centering to return mentally to math and physics – it’s a place where we’re all humble.

With that, let’s dive back in!

**Section 2.1: Displacement, Time, and Average Velocity**

A lot of this section is pretty simple, so I won’t try to complicate it by going in depth. You already know what displacement, time, and velocity are because you exist in reality.

Displacement is when something changes location. Time is… time. Velocity is the thing that uses up time to change location. Nice and simple.

The thing you need to learn here is subtle, but important. It is this: in physics, some terms you THINK you understand have a slightly different and more rigorous meaning. First off, which of these things is a scalar and which is a vector.

Fill in the blanks here. If you understood the last chapter, it should be easy: Time is a _____. Displacement is a _____. Velocity is a _____.

Did you do it?

Okay.

Time is a scalar. Why? Because it only has magnitude, not direction. As far as I know, time only moves “forward,” so we only care about how much there is, not where it’s going. If we had multidimensional time or time that could also move backward, we might make time a vector. But, this is intro physics. So, time exists and moves forward.

Displacement is a vector. When I punch you, you get displaced a certain distance in a certain direction.

Velocity is also a vector. When you move, it’s with a certain speed (displacement/time) in a certain direction.

The word “speed” is seldom used since it sometimes creates confusion. I’ve sometimes seen used the convention that speed is the magnitude of velocity. That is, speed is how fast you’re going, without concern for direction. So, for example, if you went 50 mph west, then stopped, then went 50 mph east, your velocities would be 50, 0, and -50, whereas your speeds would be 50, 0, and 50.

Remember, the reference frame here is arbitrary. I’ve simply defined one direction as the + direction and another as the – direction. By convention, on a standard graph we say that left is the negative direction and right is positive. But, we could just as easily flip that. Or, we could say left is white and right is black. The point is that in a single dimension you can move two ways – back or forth. We denote these directions with a cross (+) or a dash (-).

Now, here’s the math.

Displacement =

That is, your displacement is your second position minus your first position. In this case, we’re just talking about your position in the x direction, but you get the idea. If you wanted to add y direction, or z direction, just use the same equation on x and y. It’s simple vector subtraction.

Velocity is just how much displacement you get per unit time. You know this intuitively from driving a car. If your velocity is 60 kilometers per hour, it’ll cost you an hour to displace yourself 60 kilometers.

They also want you to know average velocity. If you can’t guess, average velocity is… the average of your velocity. In other, more precise terms, average velocity is your change in position as time changes. This makes intuitive sense, right? If you change your position by 10 meters in 10 seconds, you have a lower velocity than if you changed your position 10 meters in 2 seconds.

Here’s the math:

Average velocity =

Lastly, they wish to introduce the student to an “x-t graph.” That is, a graph with two axes, one of which is time and the other is position. This is essentially the same as the x-y graph you know from math class. The only difference is that the old y axis is now x (x meaning position) and the old x axis is now t (t meaning time). If this is a little confusing, well, tough shit. You need to learn to think of graphs as showing relations, not as showing how x changes with y. For convenience, in high school, they always use x and y. But, that’s just arbitrary – x and t make just as much sense.

Lastly, a little brush up on its:

Displacement is in length units – meters, feet, cubits

Time is in time units – seconds, minutes, hours

Velocity is in length/time units – miles per hour, kilometers per second, hogsheads per fortnight

And that’s pretty much it for section 2.1. As I said earlier, most of this stuff is pretty simple. So, I’m trying to provide some little insight and to ask you to think a little deeper about the simple concepts. I suspect this way of thinking will serve you well as we move forward.

<3,

Zach

In the spirit of thinking a little deeper, here’s a fun fact: once you really get into the nuts and bolts of relativity, time isn’t actually a scalar. Instead, time and position get lumped together as different components of a “4-vector” (in much the same way that x, y, and z get lumped together as components of a familiar “3-vector”). Just as ordinary rotations can blend (say) the x-component with the y-component, “Lorentz transformations” can blend (say) the x-component with the t-component.

And looking ahead a ways, the exact same thing happens to blend energy (classically a scalar) with momentum (a vector) to give an “energy-momentum 4-vector”. Fun stuff.

I am not sure I would say time is a scalar because it doesn’t have direction. Technically, you are right. However, when you say that, I think you are saying you can only go forward in time.

Actually, the interesting thing is that the physics doesn’t care if you go forward or backwards in time. That is, you can have a Δt be a positive or negative value and your stuff will essentially work.

So, time being plus or minus is ok. Scalars do that. Now if you said something like “meet me at Starbucks in 23 minutes west” that would be time as a vector.