Lesson Three: In which Mom drops the ball

Lesson three! You have been very patient, until now, as we have spent two lessons making graphs and tracking the positions of imaginary objects. Now, at last, we are going to use what we’ve learned to study the motion of real stuff.

The simplest and most common natural phenomenon of motion is falling. Stuff falls. Agreed?

Yes. Stuff falls. I’ve seen it.

Figure 1: Stuff falls.

Good. We will call the phenomenon of stuff falling “gravity”. Our goal is to understand gravity. But, as I have been saying, before we can explain gravity, we need to describe it precisely. We need to find a falling object and find some way to figure out it’s position, velocity, and acceleration.

Gravity

To that end, here’s a ball. Watch as I drop it:

An artist's rendition

Ball: “Thump.”

First question: did its position change?

Yes. First it was up there, now it’s down there.

Second question: did its velocity change? Was the velocity the same the whole time it was falling or did it change throughout?

It looked constant to me.

Okay, so your hypothesis is that gravity causes objects to move downwards with a constant velocity? If so, what acceleration does that correspond to?

Yes, I suppose that is my hypothesis. As for acceleration, that’s supposed to be the rate the speed changes, but the speed doesn’t change so… zero?

Correct. Zero change in velocity means zero acceleration, because that’s the definition of acceleration.

Battling hypotheses

I have a different hypothesis. I have reason to believe that the velocity actually changes as it falls. That means non-zero acceleration in the downwards direction, where the ball is moving faster at the end of its fall than at the beginning.

So how can we determine who’s right? What sort of experiment can we do?

We could drop the ball from different heights and time how long it takes.

Sure, we could do that. We could use your balcony right here and drop the ball onto the couch. What would we do with that data?

Hmm. I’m not sure.

Well, given your hypothesis, how would you expect the different times to compare?

Oh, I think I get it. If we drop the ball from half the height, it should take half the time, and if we drop it from a quarter the height, a quarter the time, right? So we can measure some different heights and see if they match that.

Good idea. It works because it actually does distinguish between my hypothesis and your hypothesis. Your constant speed hypothesis predicts that the ratios work out like you said, but in my hypothesis the ball speeds up, and so it covers the last half of the fall faster than the first. So I predict that a drop from half the height will take longer than half the time of the full drop.

Mom's hypothesis on the left, and my hypothesis on the right.

Let’s do it then! I’ll go upstairs and drop the ball and you can time it.

Great. We can measure the height with a tape measure and we can use a stopwatch on your computer.

An artist's rendition again. It's almost like someone forgot to take pictures during the lesson.

Data analysis

Alright, we probably have enough data now to be able to make some conclusions. We dropped the ball from six heights, and for each height we have the time it took to fall averaged over multiple trials. Now what should we do with this data to check your hypothesis?

Something with ratios, I know that. My prediction was that if we dropped the ball from half the height it would take half the time.

We took repeated the drop from each height three times before averaging. (Though you can see where we messed up!)

May I make a suggestion? Let’s divide all the heights by the tallest height, and then divide all the times by the time from the tallest trial. Then for each trial we can compare the time ratio and the height ratio. How should those two numbers compare if your constant velocity hypothesis is right?

They should be… the same. Oh, okay, I get it. I’ll divide the heights and you divide the times.

The ratios of heights and the ratios of times 116"/128" is 0.9, while 0.72s/0.76s is 0.94.

There! And what are our results? How do the ratios compare.

They’re not the same. And so my hypothesis must be wrong.

Right. Congratulations, being wrong in physics and being able to show it is a big accomplishment! It’s almost as good as being right, and often just as hard.

In fact, we can tell a little more. Is the ball speeding up as it falls like I predicted, or is it slowing down? As a hint, are the time ratios bigger or smaller than the height ratios?

The time ratios all seem to be bigger than the height ratios. Isn't that what you predicted? Because the ball travels across the last half of the height faster than if you just drop it from half the height, because in the first case it was speeding up before it got to half the height.

Exactly, so this is good evidence for my hypothesis, though there's a lot more to think about.

Falling

Now that we have some experimental data saying that things don’t fall at a constant velocity, let me also give you a theoretical argument as well. Which would rather have dropped on your head? This basketball dropped from an inch high, or this basketball dropped from ten feet up?

An inch, of course.

So which one was going faster?

The one dropped from higher. Oh, I see. Yeah, that’s really obvious once you say it like that. The ball that falls further goes faster, and so its speed must be changing, and so it’s accelerating.

Exactly. Isn’t it nice?

A better experiment

But we’re still missing something. Now we know that falling objects accelerate, but we don’t know how they accelerate. Does the rate of acceleration also increase as it falls, or does it stay constant?

I don’t know.

Then your homework is to devise an experiment that will let us find out.

Can you give me a hint on how to do that?

Think about our last lesson, where we took the strobe photo of the girl and were able to figure out the acceleration at different times. How did we get that?

We recorded the position of her hand at different times and then used those formulas to calculate the velocity and then the acceleration. So my experiment needs to record the position of a falling object at different times, somehow. Okay, that helps. I’ll think about it.

Good luck!

Next lesson: Data analysis is a real pain

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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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Homework Interlude 1: Please Excuse My Dear Aunt Sally

I had a big problem with my homework, but I figured it out.

Oh? What was it?

I knew I had to use those equations you gave me to fill out the tables, but it turned out I was reading them wrong.

How so?

Well, you wrote , so I took the velocity 0.3 and the position 0 and I added them to get 0.3, then multiplied by 0.1, giving me , which is my new position. Then I did it again, 0.3+0.3 times 0.1 is 0.06. And then I get 0.036, and then 0.0096, and that started to look crazy, because the position is getting smaller each time.

Yeah, it does look crazy. What did you figure out?

I asked your dad and he reminded me of PEMDAS, Please Excuse My Dear Aunt Sally.

And what is that?

It’s the order of operations. Parentheses, Exponentiation, Multiplication, Division, Addition, Subtraction. So when you wrote it means to multiply first, then add . I was doing things in the wrong order, and that’s why my answers came out crazy.

That’s right, and I guess it’s my fault for not explaining that! I’m glad you figured it out.

It’s alright. What’s funny is that once he said “Please Excuse My Dear Aunt Sally” a vague memory of it came back. I remembered the phrase but had always assumed it was a line from some forgotten song from my childhood.

Ha ha. I guess since you figured that out we can review the homework. You were supposed to repeat the exercise that we did last week and generate your own tables of position and velocity using whatever rule you wanted for acceleration. And then you were supposed to plot some position versus time graphs. Let’s see what you’ve got.

Mom's table of values for a constant acceleration

Tell me how you generated the values in this table here.

Okay, I started with this position and this velocity at the top. Then I used the formula you gave me. So I take the velocity and I multiplied it by 0.1, then I add that to the position, and that gives me the new position, 0.6. Then I used , and I multiply that by 0.1, and add it to the velocity, which gives me the new velocity, 3.5. And then I repeated that. Is that right?

It’s exactly right. Good job. Now can we look at the plots? You have two, it looks like; one with constant velocity and one with constant acceleration.

 

Mom's plots of time versus position. The one that makes a straight line has a constant velocity. The others have

 

acceleration

 

The constant velocity one looks good. But the one with constant acceleration is looking strange. Shouldn’t the accelerating curve cross the constant velocity curve because it’s going faster and will have a bigger position? See, in your table here —- Oh wait! I see what happened.

The plots viewed the conventional way, with time on the horizontal axis and position on the vertical axis.

You put time on the horizontal axis, so I was looking at it backwards, but if I flip it around like this, it looks just right. You see how it’s curving up, getting further and further away from the constant velocity one? That’s just what we want. Good job on your homework.

Thank you.

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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 and current velocity , and I’ll give your with the formula

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 . Today I’m going to use 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 by ? 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 but , 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.

All I did this time was use , so that increases by each time. That term 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 , 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 .

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 , 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.

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