Last week I made a flippant comment on Facebook about the kinetic energy a marshmallow would have were it accelerated to lightspeed.
I knew it would be giganormous. Needless to say, being a bit of a geek, I then had to go and figure out exactly what it was.
First, though, a few words of explanation. Kinetic energy is the amount of energy that something moving possesses, because it is moving. When it hits something that isn’t moving, the energy is transferred to the at-rest object. A cute toy known as a Newton’s Cradle demonstrates this in action.
The ball hits the row of stationary balls; energy is transferred through the line and the one at the end bounces off. Because the balls are steel, it’s fairly efficient. If the cradle used marshmallows, a lot of the energy would instead go into squashing the marshmallow, because it isn’t a particularly solid object (and there are other issues such as stickiness).
The thing is, kinetic energy isn’t linear, not even in the everyday world described by Sir Isaac Newton. His equation is Ke = ½MV2, where Ke is kinetic energy (in joules, an international standard measure of energy), M is mass. Since we’re using SI, that has to be grams, kilograms etc, and V is velocity – again, in SI terms, a measure of metric distance per second.
Because the equation invokes the square of the velocity, it means that kinetic energy rises sharply. If you double the velocity, you quadruple the kinetic energy. It is not linear.
The problem, as our friend Dr Einstein pointed out, is that Newton’s mechanics don’t fully describe what’s happening out on the edges of physics. In late 1905, Einstein published his theory of Special Relativity, built in part on earlier work by other physicists, which included the famous Einstein relationship equation.
You know it. But I’ll repeat it anyway: E = MC2.
The thing about this one is that everybody knows it, but few realise what it actually means. It does NOT mean that energy becomes mass at light-speed, or vice-versa. But what it DOES mean is that a little bit of mass is the equivalent of a lot of energy. That feeds into relativistic calculation of kinetic energy, which is basically the same as Newton’s at low relative velocities. That changes the closer you get to lightspeed – it’s a logarithmic curve, which means it starts off virtually flat, then tips up and goes nearly vertical as you approach C.
In point of fact, Einstein’s relationship equation actually describes is both the relativistic and the potential or ‘rest’ energy of a mass, and it can be used to derive this equation for relativistic kinetic energy alone:
Where Ke is kinetic energy, c2 is the velocity of light squared, m is mass and v is velocity (all expressed in same-unit systems such as SI). If you solve it for v = c – in other words, for something travelling at the speed of light, the answer is ‘infinity’, which is one of several reasons why you can approach lightspeed but can’t reach it.
Now the marshmallow. According to a quick Google search, a marshmallow has a typical mass of 7 grams, and let’s suppose it’s moving at 0.99 percent of the speed of light. How much kinetic energy does it have? If you take 0.99C as 2.96 x 108 metres/second, the relativistic kinetic energy in joules is 3.695 x 1016 , which equates to 3695 terajoules and is meaningless in everyday terms.
Or is it? Let me put it this way: a locomotive with a 3500hp motor, going flat out, is generating 0.000002609949550538 terajoules a second. Actually, that’s a silly comparison. This moving marshmallow is more powerful than an express locomotive!
It turns out that a marshmallow moving at 99 percent of the speed of light has 58.6 times the relativistic kinetic energy as was released by the Hiroshima bomb, which was a mere 63 terajoules. So we’re actually talking the energies of a respectable Cold War-era nuke here.
This immediately begs a lot of questions. If relativistic marshmallows got popular, who’d get the pink ones and who’d get the white ones, and would there be a difference? Would politicians worry about not having as many as the other side – a kind of ‘marshmallow gap’, like the ‘missile gap’ of the late 1950s only sillier? And would people have to set up treaties to limit the number of marshmallows each side had? I think I should probably stop there.
If you want to know more about Einstein and his stuff, and in general about how weird our universe actually is, check out my short book Explaining Our Weird Universe 1, which is a quick overview – no equations, promise!
Copyright © Matthew Wright 2016