A Leap Day 2016 Mathematics A To Z: Energy
Another of the requests I got for this A To Z was for energy. It came from Dave Kingsbury, of the A Nomad In Cyberspace blog. He was particularly intersted in how E = mc2 and how we might know that’s so. But we ended up threshing that out tolerably well in the original Any Requests post. So I’ll take the energy as my starting point again and go in a different direction.
When I was in high school, back when the world was new, our physics teacher presented the class with a problem inspired by an Indiana Jones movie. There’s a scene where Jones is faced with dropping to sure death from the rope. He cuts the bridge instead, swinging on it to the cliff face and safety. He asked: would that help any?
It’s easy to understand a person dropping the fifty feet we supposed it was. A high school physics class can do the mathematics involved and say how fast Jones would hit the water below. You don’t even need the little bit of calculus we could do then. At least if you’re willing to ignore air resistance. High school physics classes always are.
Swinging on the rope bridge, though — that’s harder. We could model it all right. We could pretend Jones was a weight on the end of a rigid pendulum. And we could describe what the forces accelerating this weight on a pendulum are going through as it swings its arc down. But we looked at the integrals we would have to work out to say how fast he would hit the cliff face. It wasn’t pretty. We had no idea how to even look up how to do these.
He spared us this work. His point in this was to revive our interest in physics by bringing in pop culture and to introduce the conservation of energy. We can ignore all these forces and positions and the path of a falling thing. We can look at the potential energy, the result of gravity, at the top of the bridge. Then look at how much less there is at the bottom. Where does that energy go? It goes into kinetic energy, increasing the momentum of the falling body. We can get what we are interested in — how fast Jones is moving at the end of his fall — with a lot less work.
Why is this less work? I doubt I can explain the deep philosophical implications of that well enough. I can point to the obvious. Forces and positions and velocities and all are vectors. They’re ordered sets of numbers. You have to keep the ordering consistent. You have to pay attention to paths. You have to keep track of the angles between, say, what direction gravity accelerates Jones, and where Jones is relative his starting point, and in what direction he’s moving. We have notation that makes all this easy to follow. But there’s a lot of work hiding behind the symbols.
Energy, though … well, that’s just a number. It’s even a constant number, if energy is conserved. We can split off a potential energy. That’s still just a number. If it changes, we can tell how much it’s changed by subtraction. We’re comfortable with that.
Mathematicians call that a scalar. That just means that it’s a real number. It happens to represent something interesting. We can relate the scalar representing potential energy to the vectors of forces that describe how things move. (Spoiler: finding the relationship involves calculus. We go from vectors to a scalar by integration. We go from the scalar to the vector by a gradient, which is a kind of vector-valued derivative.) Once we know this relationship we have two ways of describing the same stuff. We can switch to whichever one makes our work easier. This is often the scalar. Solitary numbers are just so often easier than ordered sets of numbers.
The energy, or the potential energy, of a physical system isn’t the only time we can change a vector problem into a scalar. And we can’t always do that anyway. If we have to keep track of things like air resistance or energy spent, say, melting the ice we’re staking over, then the change from vectors to a scalar loses information we can’t do without. But the trick often works. Potential energy is one of the most familiar ways this is used.
I assume Jones got through his bridge problem all right. Happens that I still haven’t seen the movies, but I have heard quite a bit about them and played the pinball game.