Some of us have talked earlier about the problem of the big four kinematics equations. In the text for my intro physics class, students are given these four equations:
Anyway, the strategy students are told is to select an equation that both has the variable they are looking for and has the other variables they do know. I sat down and thought about the logic of this approach for a while, and decided that if we really think this is what is best for students (which I don’t), we should also give them this equation:
Yes, of course, it’s ridiculous. But giving them this fifth equation is no more ridiculous that giving them the four above either. In fact, giving them the fifth at least completes the absurdity to its logical conclusion. And I’m all for that. See, given six variables, you need five equations to relate each one. Now, students will never have to worry about having to use two equations.
Take that algebra-based physics!
Teach. Brian. Teach. 
Yuck to equations. Slope and area for the win!
When we got to the projectile motion unit (which I delay until after momentum so that it’s a nice and comforting review instead of “Ahh! Why so many constant acceleration problems! Ahh!”… anyway…), I posed the challenge to my kids to come up with an equation for ∆x that doesn’t have ∆t in it and works for any constant acceleration graph (including when the initial velocity isn’t 0 m/s). If you’re drawing and using graphs, that’s the only general equation worth knowing since it keeps you from doing all of that algebra again and again each time you have a problem that’s going to require using both the slope and the area to solve. But they aren’t allowed to use it unless they’ve derived it themselves, and I don’t even pose the question to them until we’re 3 units past building the constant acceleration model.
Also, when they derive it themselves, I’ve never seen any kid write it as vf^2 =… Most write it as the difference of squares or else write it as ∆x =…
Brian,
I totally sympathize with the trouble you have of having to teach a class that is focused on equations, and I like your addition of one more equation to help students out. I wonder if you might not also add some exercises or test questions where students are to identify which terms correspond with which areas in the velocity graphs.
Yeah, I’m not actually going to give them this fifth equation… my cynicism runs only so deep. And yes, it would be nice to spend more time talking about the ideas of accumulation and rates of change, especially with respect to other representations besides equations. I’m not sure I have the time to do that well, but I do want to help them better understand more about how all of this begins and builds off our choice to describe motion with a number line and I really want to better build a concepts of constant velocity and acceleration.
I have a lot of plans for teaching the first 6 weeks of the course differently–including tweaks to labs, quizzes, feedback, and how I choose problems and structure discussion for white-boarding. I hope to be rolling out a lot of these ideas here, and I will be really interested to see.
That aside, I actually think the equation itself is interesting to think about. It’s backwards from how we usually think about it–pretend the particle had its final velocity the whole time, and then subtract off the extra it didn’t actually accumulate because it started from some initial velocity.