Peter, my brother-in-law, tells me that his son Charlie asked him a physics question the other day that he couldn’t answer: why is it E=mc-squared, and not mc-cubed or some other power? I told him I’d think about it and maybe post something on my blog. Well here goes.

For starters, this question isn’t really about relativity. It’s about energy, and E=mc^2 only makes sense if energy has the units of (mass)*(velocity)^2. So we might as well ask: why is kinetic energy defined as KE = ½*mv^2?

The crazy thing is I don’t have a really good answer to this question. I did once upon a time, but I don’t any more. That time I knew the answer was almost forty years ago when I was in high school and we had just learned the formula for kinetic energy. I remember asking why is that the kinetic energy, when all of a sudden another student declared with conviction: “Can’t you see? It’s just the indefinite integral of momentum with respect to veloctiy.” (My brother-in-law will not have a hard time guessing that the “other student” in question was none other than Randy Ellis, currently a professor of computer science in Kingston, Ontario.)

This was an amazing explanation which blew my mind. We were all (including Peter) taking calculus together, and it struck me as obvious that if you just wrote

∫mvdv

that the solution was indeed 1/2mv^2, the formula for kinetic energy.

It took me about three days to realize that this didn’t even mean anything! Integrating what for what purpose? It was just a bunch of letters on the page, and it didn’t explain anything. I still don’t know what it means. I’m not going to say you absolutely can’t make any sense of this. I’m just saying that at the high school level, it couldn’t have meant anything to any of us.

Still, there are pieces of the truth in all these things. We know that work is (force)*(distance), and we know that force is (mass)*(acceleration). Putting these together we know that the units of work, and hence energy, must at the very least be equal to kilograms*(meters)^2/(seconds)^2, which are indeed the units found in the formula for kinetic energy, and also in Einstein’s formula. So it really can’t be E=mc^3 or something else.

But just having the units line up still doesn’t completely satisfy me. Why is kinetic energy proportional to the square of the velocity? Why do we keep track of 1/2mv^2 and not some other power of v? In fact, we do keep track of another power of v, namely the first power! We keep track of mv and call it momentum, and we say that momentum is conserved. Then we keep track of 1/2mv^2 and call it energy, and we say that energy is conserved! Why don’t we keep track of 1/6mv^3 and call it something else, and look for a new conservation law? Where does it stop?

We can look for some guidance to other forms of energy, because of course it doesn’t start and stop with kinetic energy. There is, for example spring energy, with the formula ½*kx^2 where x is the displacement, and there is also the energy in a capacitor, ½*CV^2 where V is the voltage. These formulas look very much like the formula for kinetic energy. If we can explain one, perhaps we can explain all the formulas?

It turns out to be not so easy. The spring formula and the capacitor formula are indeed easy to explain, and their explanations turn out to be very similar to Randy’s calculus-style explanation of the kinetic energy formula: the only difference is that unlike kinetic energy, the calculus explanation makes perfect sense in these cases! In both these cases the energy is realy the product of two different causes: for the spring, it is the force and the displacement, and for the capacitor it is the voltage and the charge. In both cases the two variables are proportional to each other, so instead of writing (voltage)*(charge), you can just write (voltage)^2 and multiply by “capacitance”, which just hides the fact that capacitance is simply the proportionality factor which relates voltage and charge. Same with the spring constant.

I don’t know any way to apply this same argument to the formula for kinetic energy. It’s not clear to me in any way that kinetic energy must be the product of momentum and velocity in the same way that the energy in a capacitor is the product of charge and voltage.

So how do we explain it? I suppose the easiest thing is to just say that that’s how it is, it works and gives us useful results, and so we accept it. But I can’t just give up that easily. I’ve been turning this over and over and I’ve come up with an explanation. I’m not that happy with it but it's going to be the topic of my next post.

For starters, this question isn’t really about relativity. It’s about energy, and E=mc^2 only makes sense if energy has the units of (mass)*(velocity)^2. So we might as well ask: why is kinetic energy defined as KE = ½*mv^2?

The crazy thing is I don’t have a really good answer to this question. I did once upon a time, but I don’t any more. That time I knew the answer was almost forty years ago when I was in high school and we had just learned the formula for kinetic energy. I remember asking why is that the kinetic energy, when all of a sudden another student declared with conviction: “Can’t you see? It’s just the indefinite integral of momentum with respect to veloctiy.” (My brother-in-law will not have a hard time guessing that the “other student” in question was none other than Randy Ellis, currently a professor of computer science in Kingston, Ontario.)

This was an amazing explanation which blew my mind. We were all (including Peter) taking calculus together, and it struck me as obvious that if you just wrote

∫mvdv

that the solution was indeed 1/2mv^2, the formula for kinetic energy.

It took me about three days to realize that this didn’t even mean anything! Integrating what for what purpose? It was just a bunch of letters on the page, and it didn’t explain anything. I still don’t know what it means. I’m not going to say you absolutely can’t make any sense of this. I’m just saying that at the high school level, it couldn’t have meant anything to any of us.

Still, there are pieces of the truth in all these things. We know that work is (force)*(distance), and we know that force is (mass)*(acceleration). Putting these together we know that the units of work, and hence energy, must at the very least be equal to kilograms*(meters)^2/(seconds)^2, which are indeed the units found in the formula for kinetic energy, and also in Einstein’s formula. So it really can’t be E=mc^3 or something else.

But just having the units line up still doesn’t completely satisfy me. Why is kinetic energy proportional to the square of the velocity? Why do we keep track of 1/2mv^2 and not some other power of v? In fact, we do keep track of another power of v, namely the first power! We keep track of mv and call it momentum, and we say that momentum is conserved. Then we keep track of 1/2mv^2 and call it energy, and we say that energy is conserved! Why don’t we keep track of 1/6mv^3 and call it something else, and look for a new conservation law? Where does it stop?

We can look for some guidance to other forms of energy, because of course it doesn’t start and stop with kinetic energy. There is, for example spring energy, with the formula ½*kx^2 where x is the displacement, and there is also the energy in a capacitor, ½*CV^2 where V is the voltage. These formulas look very much like the formula for kinetic energy. If we can explain one, perhaps we can explain all the formulas?

It turns out to be not so easy. The spring formula and the capacitor formula are indeed easy to explain, and their explanations turn out to be very similar to Randy’s calculus-style explanation of the kinetic energy formula: the only difference is that unlike kinetic energy, the calculus explanation makes perfect sense in these cases! In both these cases the energy is realy the product of two different causes: for the spring, it is the force and the displacement, and for the capacitor it is the voltage and the charge. In both cases the two variables are proportional to each other, so instead of writing (voltage)*(charge), you can just write (voltage)^2 and multiply by “capacitance”, which just hides the fact that capacitance is simply the proportionality factor which relates voltage and charge. Same with the spring constant.

I don’t know any way to apply this same argument to the formula for kinetic energy. It’s not clear to me in any way that kinetic energy must be the product of momentum and velocity in the same way that the energy in a capacitor is the product of charge and voltage.

So how do we explain it? I suppose the easiest thing is to just say that that’s how it is, it works and gives us useful results, and so we accept it. But I can’t just give up that easily. I’ve been turning this over and over and I’ve come up with an explanation. I’m not that happy with it but it's going to be the topic of my next post.

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