When you learn physics, you begin with the basics and build on them.
Eventually, you find that you’ve moved beyond the basics and no longer can recognize where
you’ve came from. In this video, I want to talk about one of the first topics one learns in an
introductory physics class – kinetic energy – and how it eventually seems unrecognizable.
Let’s see where it takes us.
(intro music)
Before we start, I am obligated by contract to warn you that this video is way more math-y
than most. It’s not dreadful, but if you’re a math-phobe, you might have nightmares. So,
viewer discretion is advised. No plus or equal signs were harmed in the production of this
video. However, in the spirit of full disclosure, a square root pulled a muscle, but it’s expected
to make a full recovery. So, it’s all good.
The idea of kinetic energy originates from the era of Newton, although Newton himself wasn’t a fan.
Indeed, it took several decades before physicists could even agree on what kinetic energy was.
But that’s all history and there is no longer any debate on the
topic.
Kinetic energy is the energy of motion. That motion could be moving like an arrow in flight,
or it could be rotation, or vibration. However, I want to talk about just simple
translational motion.
The equation for kinetic energy is mathematically straightforward. If you have a mass m, moving at
velocity v, kinetic energy is simply one half times the mass times the velocity squared – KE
equals one half m v squared.
And you can use that equation to solve all sorts of introductory physics problems.
It’s a great tool and it served scientists well from the early 1700s until about 1905,
when this troublemaker spoke up.
Einstein was interested in the laws of physics when things traveled at near the speed of light.
He came up with an equation not for kinetic energy, but for the entire energy of an object.
I bet you’re thinking
that I meant this equation – E equals m c squared, because…well duh…Einstein…and that’s his equation,
but no – I didn’t mean that one. That’s because Einstein’s famous equation is a special case
and one that only applies to an object at rest. E equals m c squared is for an object that
isn’t moving. It’s the energy associated with the stationary mass of the object.
Now if you want to use
Einstein’s physics to talk about a moving object, you need to use a slightly different equation.
For a moving object, Einstein found that energy is equal to a thing called gamma times m c squared.
Gamma is equal to one over the square root of one minus the velocity of the object squared
divided by the speed of light squared.
Now I haven’t derived this equation, nor have I motivated where that gamma comes from. I did
that in several earlier videos, and I put links to the play list of all of them down there in
the description. They’re great fun, but they’re a diversion from the point of this video. For
this video, just take it as a given. The energy E of a moving object is simply gamma m c squared.
Now notice that I said this is the total energy,
not the kinetic energy. The total energy of a body that isn’t acting under a force is equal to
the kinetic energy plus the mass energy. So, we can do some simple math and we find that,
at least according to Einstein, the kinetic energy of an object would be the quantity
gamma minus one times m c squared.
I guess the first thing to do is to see if Einstein’s formula for kinetic
energy and the classical formula look the same. I have plotted both of them here.
We see that they disagree for velocities that are a large fraction of the speed of light.
However, as we look at lower speeds – say under ten percent the speed of light, the two graphs
look pretty much the same. This means that the equations are numerically the same at low speeds.
And it’s important to remember that ten percent
the speed of light is just slightly slower than going once around the Earth in a single second.
So, this means that the classical formula for kinetic energy is pretty good for any reasonable
speed you might encounter.
Of course, what I’ve shown you here is simply a graph of both ways to calculate kinetic energy.
How about showing how one can get from Einstein’s equation to the classical equation?
How do you do that?
Well, for this, we need to use some mathematical tricks, specifically one called the binomial
expansion. This mathematical equation was discovered in about 1655 by Isaac Newton,
although it wasn’t formally proved until 1736 by mathematician John Colson
who held the same Lucasian Chair of Mathematics at the university of Cambridge that Isaac Newton
held before him.
The binomial expansion says that the expression one plus x raised to the n power can be written
as an infinite sum of the form 1 plus n times x plus n times (n minus 1) times x squared,
divided by 2 factorial plus n times (n minus one) times (n minus two) times x cubed, divided by 3
factorial, and so on.
This equation works for any n and for all x’s that are between minus 1 and plus 1.
If you want to work out the derivation, well that’s up to you. But that’s how it goes.
So, we can look at
that gamma factor of Einstein and see how the binomial expansion can help us. One over the
square root of one minus v squared over c squared can be written as one minus v squared divided by c
squared to the minus one half power.
So, since v divided by c is less than one, that means that v squared divided by c
squared is definitely less than one. And this means we can use the binomial expansion.
We plop in minus one half for n and minus v squared over c squared for
x and do some simplification and we end up with the final answer. This is for gamma.
Remember that gamma times m c squared is the total
energy. So, we multiply this series of terms with the m c squared and we get this expression
for energy, which is m c squared plus (one half m v squared) plus (three eighths m v to the fourth,
divided by c squared) and so on.
Remember that, in relativity, energy is the rest mass energy plus the kinetic energy and that’s
what we see here. The m c squared is the rest mass energy and the rest is the kinetic energy.
Thus, to find the kinetic energy, we subtract off the m c squared, and we’re left with just
kinetic energy. And, as we expected, it starts out with the one half m v squared and then has
some other terms.
So now you might ask yourself just how big of a deal are these extra terms? We can kind of get
a sense of that by asking at what velocity the second term is big enough to change the kinetic
energy by a one percent difference from you get from the classical one half m v squared.
When you do that, you find out that you have to be going at about twelve percent the speed of light,
which is about twenty two thousand miles per second, or about thirty five thousand kilometers
per second. Ballpark, that’s about fast enough to circle the globe once in a single second.
Given that the fastest object
made by mankind is the Sparker Solar Probe, which travels at a speed of about seven miles per second
or just over eleven kilometers per second, we see why physics professors only teach
the classical equation for kinetic energy, which I remind you, is one half m v squared.
So that’s the deal. Classical physics applies
for all reasonable speeds that you will encounter, while Einstein’s equation works for all speeds,
from zero motion up to the speed of light. And, furthermore, we’ve worked out the math of how one
can connect the two. It’s cool how things all hang together. It’s almost like that Einstein
guy was smart or something.
Okay- what did you think? Lots of math to be sure, but why not? I mean…math
is the best language for precise physics, no? If you enjoyed the video, please like, subscribe,
and share. And let us know in the comment section if you want more or less math-centric videos
and we’ll take the suggestions into consideration. In the meantime,
sit back, and marvel at this wonderful result which, once again, reminds us that physics is
everything.
(outro music)