Moving along in our countdown to Newton’s birthday, we start to deal with equations that Sir Isaac never would’ve seen, because they deal with more abstract quantities than he worked with. The first and in some ways most important of these is energy:
This is the full and correct expression for the energy of a particle with mass m moving at speed v. The notion of energy traces back to Newton’s contemporary and rival Gottfried Wilhelm Leibniz, but this particular equation involves the same square-root factor as Saturday’s definition of momentum. That tells you for sure that this particular equation comes from Einstein’s theory of special relativity.
So, why is this important?
For starters, energy is important because while it’s less tangible than momentum, it gives us another way to understand the motion of objects, as a flow of energy between different systems and subsystems. While you can solve absolutely any problem involving the motion of objects using Newton’s second law, the solutions are frequently somewhat cumbersome, and may involve lengthy numerical calculations. In many cases where working with force and momentum directly is tedious and unpleasant, you can get an answer more quickly and easily by thinking about it in terms of energy.
This particular form doesn’t give you much to work with in that sense, though, being the energy of a single particle not interacting with anything else. This still tells us something important about the way the universe works, though, particularly when the speed v is zero.
If you let v be zero, then the stuff under the square root is just equal to 1, whose square root is 1. That means, the energy of a single particle that isn’t moving is not zero, but its mass multiplied by the speed of light squared (which, incidentally, is a very big number). In other words, this gives you the world’s most famous equation, E=mc2.
But what does this mean? How can something that isn’t moving contain vast amounts of energy, a quantity typically associated with motion? The short and simple answer is that something that isn’t moving through space is nevertheless moving through time, advancing into the future at one second per second. It’s that temporal motion that’s responsible for the rest energy of Einstein’s most famous equation, giving any object with mass a large amount of energy, even when it’s standing still.
When the object is moving, the total energy given by this equation is greater than when it’s standing still, and that excess energy is the kinetic energy you may have encountered in some past physics class. It may not look like the classical formula (which traces back to Leibniz), but at low energy the classical value is an excellent approximation. And as the speed approaches the speed of light, the energy of a moving particle increases dramatically, eventually becoming many times greater than the original rest energy (which, again, is huge).
This energy, in turn, can be turned into new mass, under the proper circumstances, such as a collision between protons in a giant particle accelerator. That’s how we know about all the different types of quarks and leptons and bosons, and how physicists hope to find the Higgs boson, and a whole host of other particles that might explain dark matter and other mysterious properties of the universe.
But that’s getting a little ahead of the story. For today, just take a moment to appreciate the energy of a single particle, and tomorrow, we’ll continue our countdown with the next equation of the season, and the next piece of the story of physics.