Lesson Two: In Which Mom Ruins a Family Tradition

More position, velocity, and acceleration.

Welcome back! You made it to lesson two.

Alive and kicking.

Well then let’s get going. We introduced position, velocity, and acceleration last week, but we’re going to come back to them today because they are so fundamental. Like I said last week, if we want to be able to explain the motion of things, we first need to be able to describe the motion of things precisely. To that end, we’re going to working with some “experimental data” [Editor’s note: please imagine me making air quotes with my fingers here]

Our "experimental results." Actual work done by David Hazy.

This is a strobe photo. It’s several pictures of the same girl taken at regular intervals and exposed on the same film, so we can see them all together.

So it’s snapshots, kind of like what we did last week.

Exactly. In fact, it’s just like our activity last week, but this time we’re going to go backwards. We’re going to look at the “data” here and figure out her table of positions of velocities, then figure out what rule was generating her motion.

How do we do that?

Well, first we need to pick a single point to focus on, otherwise we won’t be able to meaningfully talk about “position.” We could pick the ball, or her hand, or her foot, anything you like, as long as we can keep track of it.

Let’s do her right hand, the one we can see the whole time.

Alright, now we need a way of measuring her position. That means that for every snapshot we need a number that somehow represents where her hand is.

Could we just measure the distance of her hand from her body?

We could, but her body is always changing position too, right? That would get pretty confusing. Last week we kept track of your position by putting a number line down on the floor. Could we do something similar here?

Like just draw a line and measure her position on the paper?

Photo with a coordinate system added, and the position marked for every snapshot.

Yes, that will work. We may as well call the starting position 0, and then measure everything from there. Also, one important note: we’re going to focus only on the horizontal motion of her hand. Her hand may move up and down (like at the end) but we’ll ignore that for now. We can do this, because motion in different directions is unrelated. Now that we have fixed our coordinate system, go ahead and tell me her table of positions.

It’s 0,3.5,6,11.5 and then oh, it moves backwards, to 10.5

The position data, presented in Bic-Vision!

Right, interesting!

From position to velocity

Now I want to know the velocity of the girl’s hand at each point in time. We have a table of the positions. How can we get the velocities?

Oh, wow. Can we do that?

We can. Think about how we generated the table of positions last week.

Oh! Wait! We have this equation, . And I know all these entries in the position table. So can I use algebra to rearrange it to tell me instead of ?

I never thought I would hear you say the words “can I use algebra?” Yes, that’s right. Go ahead and try it.

Algebra! She did algebra! She started with the equation at the top and re-arranged it into the equivalent equation on the bottom.

Okay, we’ll see if I can actually do it. I know I need to get the on the other side of the equation, and also the . And I know that when we write out the equation this way it means multiplication first, then addition. So I should do the opposite. First subtract the , then divide by . So that gives me .

I have to say I’m amazed. I’m also kind of sad: no one is ever allowed to make fun of you for being bad at math again. That ruins a cherished family tradition.

Ha ha! Take that!

Well, we can still make fun of you for how you mime talking on a phone.

Figure 5: NOT how it is done, Mom!

In any case, you did exactly the right thing. What you’ve discovered here is actually the definition of velocity. It’s the ratio of the change in something’s position to the amount of time it takes to happen. The equation you wrote down is exactly the same information as the rule we introduced last week, just written in a different form. We can use this relation to go either way: last week we used velocity to get change in position, and now we’re using change in position to get velocity.

Let’s use your equation and the entries in the position table to get the entries in the velocity table:

Table of velocities. See how one goes negative at the end? What does that signify?

Now can you get the table of accelerations?

I should just do the same thing, right? Just with . So then I get

Right again.

Voila, the table of accelerations.

Graphs again

Okay, now lets turn these tables into graphs just like last week, because we just can’t get enough of graphs, right?

Never. [Ed: I suspect this was heavy sarcasm]

So we’re going to make a graph of position versus time, velocity versus time, and acceleration versus time. Why don’t you go ahead and do that.

As I draw these points should I connect them with lines?

No! … Sorry, that was knee-jerk reaction. I always say no because visually that would imply that we know what happened in the space in between the two snapshots, but we really don’t. Maybe she moved smoothly, or maybe she did something really fast with her hand that doesn’t show up! We can’t pretend that we know. However, you can draw an approximate curve through the points. That helps guide our eye and doesn’t seem as definite.

Our beautiful tables turned into even more beautiful graphs. One each for position, velocity, and acceleration all versus time.

 These graphs are good, but we’re running low on time. I had some questions I wanted to ask you about them, but I think I’ll leave it for homework.

What did we learn?

Before we finish, why don’t you tell me what you think you learned today.

The first thing I learned is that you shouldn’t connect points on a graph.

Ha ha, good!

The main thing is that I was surprised by is how much more there was to the motion of the girl’s hand than I initially thought. Just looking at the picture I wouldn’t have thought that her hand moved backwards. I only saw that when we made the table.

And now that I look closely, I can see that the other parts of her move very differently. Like here, her foot doesn’t move backwards at all. I would have thought that everything was the same.

That’s a good insight, and it’s a big part of why we’ve spent two days working on these concepts of position, velocity, and acceleration. If we’re going to explain what happens in the physical world we first need to describe what happens in the physical world, and as you’ve noticed it’s very hard to do that without precise measurement. Now that we have those tools we can start looking at real physics. Next week, I promise.

Homework

Finally, homework. I hope that looking at the tables and graphs helped you get a sense of how position, velocity, and acceleration relate to each other.

I want you to do two things. First, use what you’ve learned to make your own strobe photos. Or rather, to draw some pictures that are kind of like strobe photos. I want you to draw a series of snapshots of a ball in four different situations:

1. Draw a ball with a positive position and a positive velocity.
2. Draw a ball with a negative position and a positive velocity.
3. Draw a ball with positive velocity and negative acceleration.
4. Draw a ball with negative velocity and negative acceleration.

Second, I want you to look a little more carefully at the graphs we made and the relationships between them.

1. When is the position curve at its highest point? What part of the picture does that correspond to?
2. When is the velocity curve at its highest point? What part of the picture does that correspond to? What is the position curve doing at that point?
3. How could you tell where the highest velocity occurs looking only at the position graph?
4. Where does the biggest acceleration occur? What is the velocity curve doing at that point?

Next lesson: Mom drops the ball

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