Note: This post is mostly inspired by conversations with Mark Hammond, an incredible teacher who has done wonders for his students by replacing their trigonometric crutches with rulers and protractors.

Vectors are mysterious things to kids—how in the world can 2N + 2N be anything between 0N and 4N? And free body diagrams are almost as mysterious—you mean I can’t tell where an object is, or how fast it’s going from a FBD? All I can know is if the velocity is changing? How useful is that? (You’ll see, I say).

I used to teach this stuff by dropping down x- and y- axes from almost the second that I put up a FBD. But then, I watch as kids struggle, not just with mastering the trig of calculating components, but with understanding the reason for choosing coordinate systems the way we do.

Student: I can get that the x-axis should horizontal when the book is on the table, but why should I tilt it just because the book is on the a ramp? And if the book is going around a banked curve, it shouldn’t be tilted anymore?
Whhhaaat?

Sure, I know there are lots of rules you can give the kids to make all this easier “Always set your axes to point in the direction of acceleration, choose axes to line up along the greatest number of forces, blah, blah” but none of these rules give them much intuition into what why you are making these choices, or what vectors really are.

Here’s the best solution I’ve come up with so far, again, with huge thanks to Mark Hammond. We start by drawing free body sketches. The point of this exercise is to practice identifying forces, especially hard to recognize ones like the force of the outside air pushing on the suction cup, or the upward frictional force holding the book up when it is pressed into the wall. We’re also practicing being very picky about naming forces fully to indicate the type of force (gravitational, elastic, etc) and the two objects that form the interaction (the gravitational force of the earth on you). Being very dedicated about this pays HUGE dividends when it comes time for Newton’s 3rd Law.

From there, we do a lab where we develop a way to measure the gravitational force on an object, and the elastic force of a spring. They develop two, simple empirical laws that allow us to know the size of the gravitational force on any object, and the size of any elastic force, if we know the spring constant and the stretch of the spring.

Now, it’s time to break out the rulers and protractors. We go back to a similar set of free body diagram questions, but this time, I tell them the mass of the box is 5 kg, and they need to find out any additional forces in the problem. This is all pretty easy stuff when the box is at rest on a flat surface (or sliding along without friction—they’re starting to get N1 down now), but it becomes much harder when the block is at rest on an incline. How do you know how big the normal and frictional forces are? I just let the kids wrestle for this a while, reminding them that Normal and Frictional mean perpendicular to the surface and parallel to the surface.

A box is at rest on an incline. How do you know which FBD is right?

Here’s where we bring back the basic idea of vector addition we came up with earlier in the week. Students know the three vectors, call them , a, b and c, must add to zero. Let’s call the gravitational force, c. This tells them that the other two forces a and b, (normal and gravitational force) must add up to the opposite of c. Then I remind them again about how the normal force and frictional force are perpendicular, and tell them it’s a little like being in manhattan.

Imagine you’re at the 7th ave subway station, and you want to get to the Carnegie Deli, due north of your position. But you can’t walk straight through a building. Instead, you have to stay on streets (frictional forces) and avenues (normal forces). How will you get there? Perhaps a picture will show it better.

And now a light suddenly goes off. There’s really only one way to get there. Walk 1/4 a block to the left on 53rd and 1.2 blocks up 5th ave. Wham-o! You just discovered vector addition, all for yourself. Now you see why 2.3 N+4.5 N = 5N (almost), and you’ll never be the same again.

Bingo! The frictional and normal forces are uniquely determined.

From this, students explore a whole host of avenues. It turns out that all this stuff follows geometry perfectly. $F_n^2+F_f^2=F_g^2$ and the fact that you know an angle, a side and another angle uniquely determines this little triangle. But we go even further. What happens to these forces if you tilt the ramp higher? The students quickly see that the normal force must get smaller, while the frictional force must get bigger, and they reason this can’t go on forever, so the box must slip.

Notice also that the streets and avenues is also setting them up for components when the time comes, but they aren’t focused on that yet. And here’s were it can get really powerful. Three killer FBDs, below:

Three similar, but very different FBDs

Normally, kids see the third free body diagram (centripetal motion) not long after they’ve just started to learn how to do full blown N2 Analysis using components, and it throws them for a loop. Here’s the same block, on the same plane, but this time, because it’s somehow moving, students are just supposed to know that they should use a horizontal coordinate system?! And since most kids just sketch FBDs, they really can’t tell a difference between the two FBDs on the right. But all of this goes away if you just have them focus on drawing the forces correctly and adding them graphically, coordinate free. Moreover, they can see some really significant insights, like the fact that the net force is greater for the banked turn than it is for the block sliding down the plane.

When they get their triangles, they’re also capable of determining the net force graphically, and then using simple geometry to find the size of these forces. Which leads to the powerful conclusion that the net force on the car going around the turn is larger than the net force on the car rolling down the incline for any angle of incline.

Net force for each of the free body diagrams above.

So I think this leads to a much more intuitive understanding of vectors, adding forces and free body diagrams than kids would get with 2 days of vector math (there was no way my kids could keep vector notation, addition and subtraction straight with such a short exposure), followed by 2 days of FBDs with components like I used to do. Free body diagrams become really useful, and not just tiny little pictures students draw to make physics teachers happy. Also, I save vector subtraction until we get to acceleration, and we do that graphically too, which pays huge dividends I’ll write about later.

And really, we don’t do components until after all the kids have drawn dozens of FBDs with rulers and protractors and someone says, “isn’t there an easier way to do this without all the protractors?” and then I’ll say, remember the streets and avenues? And suddenly, math will be useful and maybe fun.

September 18, 2010 11:54 pm

What I like about vectors is how it helps kids transition from concrete to abstract thinking. The issues you raise are beyond my expertise, but I used vector diagrams with 7th graders of static objects. The scaffolding more abstract thinking really helped them be able to think more critically about newtons laws rather than simply accept my word as truth. Glad you are wrestling with these issues!

• September 19, 2010 12:06 am

Yes! Exactly. I actually love the way the book Force Motion Energy handled vectors as a 8/9 physical science text. Actually, I loved everything about that book. Now this text has been absorbed into the equally awesome IPS text, IPS 9. My point is that you can do a lot with vectors if you aren’t forcing kids to put down axes, break them down into components and jump into algebra. Just have them look at the diagrams, and begin to use geometric reasoning to make sense of them.

A funny story that my former colleague Mark used to tell about kids coming into our physics course. We’d always have one or two kids who would blow the doors off of the pre-test Force Concept Inventory (a diagnostic test used to assess whether a kid understands Newtonian thinking), and invariably, when we would question them about what they did in middle school science (we were a boarding school, so kids came from all sorts of backgrounds), they would say something about using a really thin book, doing experiments all the time, working really hard, and “I think it was called IPS or something.”

2. March 20, 2011 8:39 am

This is how I was taught vectors in physics. First with rules and protractors, then with law of sines and law of cosines. Components weren’t introduced until way later. After we already grounded ourselves with adding displacements, addition of velocities, forces, and momentum with more geometric approaches… I have a story about this. Maybe I’ll make a post.