Teaching Physics To My Mom: Lesson One: In which Mom pretends to be a particle

The coordinate system

Welcome to physics!

I thought we would start with a game, which I’ll explain after we get set up. The first thing we need to do is to mark out a ruled measuring line on the floor. I used pieces of masking tape placed 50 centimeters apart on our living room floor, which Mom then labelled with a marker.

We need a range of 2 or 3 meters. We’ll choose one of the pieces in the middle to label 0, and then the piece to the right of that will be labelled 0.5 meters, then 1 meter, and so on like a ruler.

Mom: Wait, but what do I do with the ones to the left?

We’ll use negative numbers. So 1 meter to the left of 0 is labelled -1m and 1 meter to the right is labelled +1m.

So it’s like a number line.

Mom marking out out coordinate system on the floor.

Exactly. Positive or negative just points out different directions. Here, let’s try it out with the first stage of the game: You’re a particle, and I’m The Laws of Nature. You move around on the number line according to the rules I give you. We’ll take it in steps. For right now, we’ll just practice; I’ll tell you a position and you move to the corresponding spot on the floor. Ready?

I guess so.

First, start at zero. Here? Good. Now move to 1 meter. Now move to 0.5 meters. [Good again.] Now move to -1 meter.

That means I move over here, to the left of zero?

Yep, you got it. Now move to -1.5 meters. [Mom moves to the right spot.] Now pay close attention. You moved from -1 to -1.5 Did you get closer or further from zero? Further. And which direction did you move? Left. Okay, let’s try one more thing. Increase your position by 2 meters.

Oh, hmm. Okay, 0.5, 0.5, 0.5, 0.5. That makes 2 meters. Now I’m at 0.5 meters.

Correct. Now change your position by -1 meters.

Uh, I guess I go left? 0.5, 0.5, now I’m at -0.5 meters.

Mom at 0.5 meters, doing what she called a “particle dance.”

Yes! Good job. You can change your position by positive or negative numbers, and all it means is what direction you’re moving.

If you’re wondering why we’re doing this, it’s because this is the most fundamental task in physics. The job of physics is to explain why objects move the way they do. But objects do all sorts of interesting things, so the first thing we need is a way to describe their motion precisely. The solution is a coordinate system, which is where we assign every possible location a number, called its position. At every point in time, every particle has a position that we can keep track of, at least in principle. At a certain level, this game of “move around on the number line” is all physicists do.

The Physics Simulator

Okay, now the game is going to get a little more complicated. Here are the rules. You’re going to start at a point on the coordinate system that I tell you, and you’re going to take this pad of paper and keep a table of two things, your position, and something new called your velocity. At every step you’re going to tell me, The Laws of Nature, your position and velocity. Then I will tell you to move to a new position and give you a new velocity, and we’ll start over at the next step.

But here’s the trick, I’m not going to tell you random numbers for your new positions and velocities, I’m going to be following a rule myself. The rules use two formulas. So every step you’ll give me your current position $x_{\text{current}}$ and current velocity $v_{\text{current}}$ , and I’ll give your $x_{\text{new}}$ with the formula

$x_{\text{new}} = x_{\text{current}} + v_{\text{current}}\times \Delta t$

and then—-

Hold on, what’s that little triangle thing?

Ah, well we’re pretending that you’re a particle moving along and we’re taking a snapshot of your position at regular intervals. The length of that imaginary interval is $\Delta t$ . Today I’m going to use $\Delta t = 0.1$ seconds.

Oh, doesn’t your brother-in-law take snapshots of particles with his cyclotron? Is that like what we’re doing here?

I guess so. I don’t really know what he’s been working on lately. But it sounds kind of like what we’re doing right now, yeah.

It’s funny how you two don’t talk more about this stuff, you’re both physicists after all. It’s weird.

Granted! But back to our game—

Wait, I’m confused again. Why do we multiply $v_{\text{current}}$ by $\Delta t$ ? Why don’t we add?

Good question. See, velocity is the rate of change of your position in time. If you’re moving along at some velocity for an amount of time, you’re going to go a certain distance. How far you go depends on both your velocity and your time. If you are going 5 meters per second for one second, you’ll travel 5 meters. But if you go 5 meters per second for 0 seconds you’ll go 0 meters. We need multiplication because $5\times 0 = 0$ but $5+0 = 5$ , which would give us the wrong result for our purposes here. It’s also because velocity is defined as the ratio of ——

No, this isn’t working. This is like when I learn a new board game and everyone wants to explain the rules to me. I always get frustrated because I just want to start playing and figure it out as I go. Can we just start the game?

Yes, I think we should. Go stand at -1.5 meters, and note down that your starting velocity is 3.

Starting the game

Now I’ve got my paper and my calculator and we can start. Tell me your position and velocity. -1.5 and 3. Good, that makes your new position and velocity -1.2 and 3.

Okay, I’ve moved to -1.2 and my velocity is 3.

Then your new position is -0.9 and your new velocity is 3. Got it, -0.9 and 3. Now with that position and velocity you should move to -0.6 and keep velocity 3.

The table of positions and velocities for the first try. Note the author’s inability to draw a straight line.

I think I see the pattern in the velocity. It just stays at 3 every time.

Correct. What about the pattern in the position?

I don’t see that yet. Wait! Oh! It changes by positive 0.3 every time!

Yeah, you’ve got it. So what’s the next position and velocity I’m going to give you?

It must be -0.3 and 3.

Yes! Good job. You just did physics! You observed some data, noticed a pattern, formulated a rule that possibly explains the pattern, and then made a prediction using the rule. That’s basically all of physics right there.

This pattern is called “constant velocity.” The rate at which you change your position doesn’t change, so when we look in steps of equal time, you always move the same distance.

Constant acceleration?

Okay, that rule was pretty easy. Now let’s start over and I’m going to use a new rule. Start at -1.5 again, but with velocity zero. [Editor’s note: this time I’ll just list out the positions and velocities for each step in a table. See if you can see the pattern yourself]

The positions and velocities with the new rule. What’s the pattern?

Oh! I think I see it! So my next velocity is 0.9, and my next position is 0.9. No, wait, 0.75

Yes! That’s it! Good job. Now, can you explain to me the rule you used to get that?

The velocity increases by 1.5 every time. For the position, you take the position in the previous row and add the velocity from that row, but multiplied by 0.1

Perfect. That’s the rule I’ve been using. In fact, if you write down what you just said with mathematical language, you get the formulas I tried to show you earlier. Let’s go back to those now that you’ve figured it out yourself.

$x_{\text{new}} = x_{\text{current}} + v_{\text{\current}}\times 0.1$
$v_{\text{new}} = v_{\text{current}} + a\times 0.1$

All I did this time was use $a = 15$ , so that $v$ increases by $1.5$ each time. That term $a$ is called “acceleration.” Acceleration is the rate of change of the velocity, just like velocity is the rate of change of position. A big acceleration means your velocity changes a lot every time step. And notice that if we take $a=0$ , then there’s no acceleration, the velocity is constant, and we get the rule for the first round we played. Both rules have the same form, just with a different value of $a$ .

Graphs and prediction

Okay, there’s one more thing I want to do, but it involves graphs.

Bring it on.

Alright, then let’s do it. We’re going to make a graph of your position versus time. So we draw time on this horizontal axis, and I’ll mark out the specific times we looked at: 0, 0.1, 0.2, 0.3, etc. Then I put position on the vertical axis. Remember it can be positive or negative. Now we look at the tables you made and put the entries on the graph as points. Let’s do the constant velocity one first.

The “constant velocity” position-versus-time graph in blue, and the “constant acceleration” one in red. Luke accepts all responsibility for the mess.

Now I have a question: looking at the blue curve, if this pattern continued, where would you end up after a hundred steps, approximately?

Way over in your neighbor’s house, right?

Right. Good, you can extrapolate the pattern on the graph. Now let’s put the constant acceleration data on here so we can compare. Same question, but with the red curve: where would you end up after one hundred steps?

Still in the neighbor’s house.

Which rule would take you further?

I guess the constant acceleration rule because my speed keeps increasing. This graph keeps curving up but the other one just goes up in a straight line.*

Right. If the first rule takes you in the neighbor’s house after one hundred steps the second rule will take you all the way down the street.

This is another key lesson! All we did was change the acceleration and we got vastly different behavior from our particles. We’ll see this over and over in physics. Small variations in the rules can produces remarkably varied results.

Enough for now, plus some rambling.

This is probably a good place to stop for the night, but first I’ve got to give you your homework.

Can I ask a tangential question first? We’ve been talking about particles, and you’ve mentioned that everything is made up of particles moving around, but what exactly is a particle? Are they atoms? Did Newton know about particles?*

Sure, good question. Unfortunately “particle” is a word that means different things depending on what kind of physics you’re talking about. The way I’m using “particle” now is very different from the way I use it at work, where I’m a theoretical particle physicist, which Newton would have known nothing about.

For our purposes, a particle is an object that doesn’t break apart when you push it. That includes a lot of things! An atom is a particle, but so is a baseball, or this chair, or the moon. On the other hand, water is not a particle, because it ripples and breaks apart when you try to move it. Though, it turns out if you model water as being made up of lots of little particles you can do very well at predicting its behavior. That sort of thing is the first sign that people had that everything is made up of little particles.

And one more thing, if the chair is made up of tiny particles, what makes it a chair? Why doesn’t my hand just go through it?

It’s all about the rules the particles follow. We just saw how different rules can result in very different behavior. The particles that make up the chair are following a rule that makes them want to stick together, so they resist when you press on them. The particles in water are following a different kind of rule, so they do move out of the way when you put your hand in.

Huh, well that answers a lot of the debates my students used to have in my Theory of Knowledge class.*

Ha. Glad I could help.

Homework

You did an excellent job today figuring out the rules that you were following. Your homework is to apply the rules yourself. I’ll write down the formulas for you, and you need to use them to generate a table and a graph for the motion of some particle. You can use any rule you like for the acceleration. You might want to start simple, like with constant $a$ , but in principle there are no limits. You can show me your table and graph when we meet next week.

Okay, I’ll try!

Next post: Lesson 2 - More on position, velocity, and acceleration.

Teaching Physics To My Mom

Friday, February 17, 2017

Lesson One: In which Mom pretends to be a particle