Physics Curriculum Question

I am thinking about re-organizing the first half of my physics classes to push much of kinematics until later — maybe much later! Here is what I may go with:

Start with vectors…displacements, forces, velocities (and for now, velocity is just a fancy word for speed)

This would include vector addition, vector components, an intro to the essential right-triangle trig, and applications including “river crossing” and equilibrium problems (hanging signs and such).

Next, a minimal introduction to kinematics…

This would include a definition of average velocity, and an examination of x vs t and v vs t graphs for constant positive velocity. Then, a look at constant acceleration: its x vs t graph, its v vs t graph and the definition delta V over delta t. BUT NOTHING YET ABOUT: area under v vs t, kinematic formulas, word problems, positive vs negative velocity and acceleration…

QUESTION #1 FOR MY COLLEAGUES: Is that enough kinematics to take on Newton’s laws? Am I forgetting something?

On to Newton’s Laws…

It seems crazy to me how long it can take me to get to these if I do a full treatment of kinematics (and even projectiles!) before Newton’s laws. If I adopt this new arrangement, I think I can start Newton’s laws in September. This does mean that I can’t spiral F=ma problems which require kinematics to get the acceleration. But eventually I can spiral the other way: put F=ma into kinematics problems.

On to…well, I am not sure…

At first, I was thinking that next I would go back and finish up kinematics and projectiles. Then, on to work and energy, followed by momentum. (I know others prefer to reverse those two…I am not religious about that.)

But then I got to wondering: could kinematics and projectiles even wait until after all of this? I might have to skip the derivation of W = delta KE but that feels minor. Could kinematics and projectiles come right before circular motion? From projectiles right into circular motion feels kind of natural.

So that leads to QUESTION #2 FOR MY COLLEAGUES: If you delayed kinematics in the way I am considering, when would you go back to it?

I don’t want to open comments on this blog (too much weird spam) so please find me on twitter @pckeller if you would like to chime in. Thank you!

Enough Calculus to Get Started

A brief  introduction for AP Physics C Students

And anyone else who wants an overview of what we do when we “do calculus”

AP Physics C is a calculus-based course.  At my school, calculus is a co-requisite.  It would be good if we could use the language of calculus right from the start.  But most of the students have only seen a little calculus so far (or none at all).  So over the last few years, I have written these blog posts to try to get them off to a running start.  

Also, while it was not my intention, I think these posts would help anyone heading into their first calculus class to get an advance overview of what lies ahead.

Note to my students:  For some of you, most of this will be a review – but definitely not all of it!  But for others, this will all be new. Just take your time, read slowly, and feel free to ask lots of questions — you can save them for September or email me as you think of them! Or post them in our Google Classroom (which is already open).

One other note:  we will be talking a lot about rotational physics this year.  It would be helpful to have a clear understanding of radian measure.  If you feel at all shaky, you should review these two posts:  Angle Measurement for Pizza Crust Lovers and That Radian Feeling.

I hope you have a relaxing summer and that you are looking forward to another year of interesting physics.

— Mr. K

POST  TOPIC
1 Hairy Questions Slope and Rate of Change
2 Even Hairier Acceleration
3 Velocity NOW! Instantaneous Rate of Change
4 Slope-Finding Functions What a derivative tells us
5 Derivatives: The Building Blocks Some often-used derivatives
6 Derivatives: Combining Functions Sums, differences, and more
7 Simple Version of the Chain Rule Composition of Functions
8 Making the Best of Things Optimization
9 Exponential Decay – Part 1 1st Order Diff. Eqs.
10Exponential Decay – Part 2  Understanding time constants
11 A Magical Theorem, Part I Anti-derivatives vs. Areas
12  A Magical Theorem, Part II Areas via Anti-derivatives
13Part III: Areas the Hard WayRiemann Rectangles
14Part IV: Areas in Physics“Area” under a graph
15Part V: Adding Up the PiecesApplying the Magic Theorem

Part V: Putting the Pieces Together

So many pieces!

Finally, finally we have arrived at the point all of these posts have been building toward:

In physics, we only occasionally care about “area”.  But we often care about adding up an infinite number of tiny little things.  And when we do, if we think of them as representing areas, we can use our Magic Theorem to add them.

I want to demonstrate this using as little physics as possible.  So this next example may seem a bit contrived.  You will see more significant examples as our year goes on (perhaps more in E & M than in Mechanics). But this is a good one to begin with:

A beam is 2 meters long. It has a density that varies along its length. As a function of distance from the left end, the linear density (in kg/m) is given by the function:

Find the mass of the beam.

Before we jump in, let me say a little about what we are hoping to do.

We have a quantity that varies continuously: in this case, it’s the density.  (Also, please do not be intimidated by the fact that we are using the Greek letter, lambda, to represent the density function.  I think we do this because it is a LINEAR density (mass/length) and so we want a letter that begins with L.)

If we chop our beam into short pieces, we can pretend the density is constant for a given piece. Really, density does vary but if the piece is small, density doesn’t change much. And we are eventually going to make the pieces reeeeally small.  This is pretty much the same thing we did last time, when we pretended velocity was constant over a time interval.

We then multiply the length of each piece by its density to find the mass of that piece and then we add them up.

So here’s the beam, nicely chopped:

We focus on one individual piece, say that blue one, and find its mass:

And we add up all those little masses:

As we have before, we can improve our estimate by chopping into more pieces – or even better, an infinite number of pieces:

Now let’s really examine that expression.  It sure looks a lot like a Riemann sum!  It looks like you are getting ready to find the area under the curve:  f(x)=1+x-x2/2 by breaking it into rectangles.  And for that we can use the Magic Theorem. 

We can find the anti-derivative ourselves or use Wolfram/TI/whatever to get:

And then see how much it changes:

We are done! And yes, that is also the area under that graph, but that’s not really the point.  We were not interested in finding an area. We just wanted to know the mass.  But as we have just seen, the same method that lets us find areas also helps us to add up the little pieces!

Well, that’s all I wanted to say about the Fundamental Theorem and how it is used in physics.  If you have stayed with me through this entire 5-part journey, thank you so much for your persistence.  I find this to be pretty challenging stuff.  I will be happy to help you through it again when we encounter this during our year in AP Physics.  I hope the time you spent working through these posts makes it easier for you when you see it again.

See you soon!

Mr K

Back to Part IV

Go to Summer Reading Outline

Part IV – Areas in Physics

Here is an easy warm-up from your first year of physics class:

A car moves at a constant velocity of 6 m/s for 5 seconds. How far does it travel during that time?

When velocity is constant, you can multiply the velocity by the time and get the displacement:  

But it is interesting to look at a velocity vs. time graph that models this situation:

The area between the graph and the t-axis forms a rectangle.  So when you are multiplying the velocity (6 m/s)  by the time (5 s) you are also finding the area of that rectangle!

But is this really an area? Or is it more of an “area”? Look at the lengths of the sides and note the units. We expect the lengths of the sides of a rectangle to be measured in units of distance. But this one has one side measured in velocity units (m/s) and the other in units of time (s). And when you multiply them, you get a unit of distance, not a typical unit of area. So the “area” on the graph represents some other physical quantity we care about.

Now here’s something your physics teacher might ask THIS year:

A car’s velocity increases with time according to the function  v(t)=t2. How far does the car travel in the time from t = 0 to t = 5 s ?

Since we know how to find displacement when the velocity is constant, let’s adapt the idea we used in math class to generate an approximate answer.

Here is our plan:

  1. Divide the time into intervals, each 1 second long.
  2. Use the formula v(t)=t2to find the velocities at the beginning of these intervals. 
  3. Use those velocities to find the displacements for each one-second interval and then add them up to find our approximate answer.

So you are pretending that instead of gradually increasing, the velocity stayed constant for 1 second at a time, jumping to the next value at the beginning of each second (depending on whether we want to over-estimate or under estimate our answer). 

Why don’t we graph what thAT looks like on a velocity vs time graph?

“It’s like Déjà vu all over again” – Yogi Berra

I hope this looks familiar!  It’s just like what we saw in the last post.  Treating the motion as if if were a series of constant-velocity intervals is just like approximating the area under a curved boundary by using a bunch of rectangles!

I used a spreadsheet and put the results of these calculations in the chart below. Take a look at the overestimate and the underestimate. You might recognize those numbers…but once you understand what we are doing here, you won’t have to do these calculations at all!

So now we can say that the cart travels somewhere between 30 and 55 meters. You can probably guess what your teacher will say:

“I like your method, but I wish it were more precise.  Can you divide the interval into more, finer intervals?  Like maybe 10 of them? Or even 50? 

[It’s amazing how your math teacher and physics teacher are so in tune.  It’s almost as if they were one person…]

Here’s what 50 rectangles look like (again). Seriously, the only difference is that I labeled the x and y axes to be velocity and time.

And if we want the exact answer, we evaluate the limit:

Except we still don’t have to do it this way! We can still use the Magic Theorem! Really, all that has changed is that we are calling the function v(t) instead of f(x):

It is true that we have just found an area under a graph.  But what we also found, and what we actually wanted to find, is the answer to a physics question: how far did the car move?  It is just a helpful connection that we can think of that as an area.

IT’S NOT JUST FOR DISPLACEMENT

Here are some other things we will calculate this year. In each case, we can think of it as an area and then use the Magic Theorem.

1. Impulse exerted by a force that varies with time:

This is also the area under the Force vs time graph.

2. Charge delivered by a current that varies with time:

This is also the area under the current vs time graph.

3. Work done by a force that varies with position:

This is also the area under the force vs. position graph.

Once you are comfortable with this, you can find these things without even drawing the graphs! Integrating will come to mean: adding up infinitely many tiny contributions to an overall total.  And as you will see in the next (and last!) post in this series, that’s another powerful way use integration in physics.

Ahead to Part V

Back to Part III

Go to Summer Reading Outline

Finding Area the Hard Way

The Magic Theorem, Part III

The previous two posts have wrestled with the relationship between an area (the “definite integral”) and antiderivative (the “indefinite integral”).    In short, they are related by the Fundamental Theorem, or as I have been calling it, the Magic Theorem:

You can find the area under a graph by finding how much its antiderivative has changed.

In mathematical notation:

But that still leaves us with the question:

Why do we care about area in physics?

I think the answer to that question is that really, we very rarely care about the literal “area” as in area of some planar region, measured in square meters.  But we frequently care about other physical values that we can represent as areas. And the methods we use to find those areas can be extended to other useful things.

To explain this, in this post we will step back and look at how you can find area in math class before you learn the Magic Theorem.

[Just for extra emphasis, I will say again: what follows is not a method you will need in AP Physics.  In fact, except for a few days, you won’t even use this method in math class either.  But if you want to understand what the Magic Theorem does for you, you have to understand what it replaces.]

 In math class one day, your teacher might ask you to find the area between the x-axis and the function f(x)=x2:

Now maybe rectangle areas are the only ones you know:  

area = length times width 

You are kind of stuck. But  your teacher asks you to keep trying and says they are willing to accept an approximate answer.  Oooh.  Now there is a way forward here.

You could slice the region into a handful of rectangles. Let’s start with five of them:

You could then calculate the area of those rectangles and add them up to get a rough idea of the area of the entire region.  [Notice, I say we COULD do this.  It’s a tedious task to do by hand, and in the end we won’t have to do it. But we could if we wanted to. As you can see, I let Desmos do the calculations for us.]

In the diagram above, the rectangles are INSIDE the region, so we have underestimated the area. We could have chosen to put them OUTSIDE, thus overestimating:

So we can see that the area we are looking for is more than 30 and less than 55.  That’s an approximation, but it’s better than no knowledge at all.  It might even satisfy your math teacher for a while. But then they come back to you with a request: 

“I like your method, but I wish it were more precise.  Can you divide your graph into more, finer intervals?  Like maybe 10 of them? Or even 50? 

Well, you can do this too. Here are 10 “outside” rectangles:

And here, side by side, we have 50 “inside” rectangles and 50 “outside”:

We can now make two observations about what happens when you use lots of rectangles:

1. The difference between “inside” and “outside” estimates becomes smaller and smaller.

2. The amount that you are overestimating or underestimating by also gets smaller and smaller.

If we could some how let the number of rectangles approach infinity, we would know the true area of this region.  In fact, that limit is the definition of the area of this region. 

This way of finding area is called the Method of Reimann Sums.  The area is often expressed using summation notation:

Setting up these expressions and then calculating the limit as the number of rectangles approaches infinity can be quite daunting. Mastering this method is a rite of passage in calculus class and I admit that it is possible you will not enjoy the process.  It is good to understand what Riemann Sums are, but it is also really nice to not have to use them.  That’s what is so magical about the Fundamental Theorem — it is so much easier that it feels like magic. Just to remind you of what we do:

1. We find a function which has x2 as its derivative.  For example:

2. According to our magic theorem, the area we want is the change in that antiderivative:

That’s not so bad! But also, now we have all the pieces in place to answer the question that started us down this road: why do we care about area in physics? 

Onward to Part IV of our adventure!

Ahead to Part IV

Back to Part II

Go to Summer Reading Outline