In physics the equation for momentum is this p equals mv. So if you've got a certain mass moving at a certain velocity it's got a momentum equal to the mass times the velocity. Like any equation you can draw this as a graph.
I've chosen the axes to be momentum and the velocity and the mass gives the gradient or slope of the line. Now let's say we want to work out what the area under the graph is. This is simply the area of a triangle, half the base times the height. And by the way, this is true no matter what the gradient is.
So the area is a half times P times V. Now we can substitute in the original equation for P, because P equals mV. So that gives the area to be half times m times V times V, or a half mV squared. Now you may be familiar with this equation already.
It's the kinetic energy of a moving body. And the mathematics we just did shows how these two equations are related to each other. And in physics, we call this a derivation. A derivation is basically how you invent new physics. You take the knowledge that you already have and you combine it in a new way to come up with a description of something new.
And up to university level, derivations involve two bits of mathematics. The first is algebra and the second is calculus. Algebra is basically taking equations...
rearranging them or putting them together kind of like when we substituted p equals mv in the example beforehand calculus involves two parts integration and differentiation integration is when you're finding the areas underneath curves and differentiation is when you're finding the gradients of curves and the job of calculus is basically to describe how certain things change when other things change And this happens all the time in physics, so calculus is an incredibly useful mathematical tool. Now we didn't actually do any calculus in my example because the graph is a straight line, which basically means it's easy enough to use simple geometry to find the area, and calculating the gradient of a straight line is simple. It's just the height divided by the width of the triangle, which in my example is the mass, m. Things get more difficult when you've got an equation that's got a curved line, like the plot of kinetic energy that we've derived. It's a curved line because it's got a v squared in it.
Now you can't just use a simple triangle to find the area and gradient, now you need to use integration and differentiation. Now I don't have time to teach you all of calculus in this video but I'll give you a brief description of the way it works. The idea behind integration is that you start by approximating the area under a plot with a series of rectangles. So we can write that each area is a half mv squared, which gives the height, times a small width, which we'll call delta v. And the total area is the sum of all of these rectangles. The trick to integration is that as you reduce the width of these rectangles to be smaller and smaller, this approximated area gets more and more precise and becomes perfect.
when the width of the rectangles gets infinitely small. Then we rewrite this sum as an integral which looks like this. If we do this integration we come up with this equation and this is already kind of interesting because as physicists we don't have a name that describes the quantity that we've just found, so we're already sort of in the wilderness of physics. A physicist's description of this would be the integral of kinetic energy with respect to velocity.
But seeing as we've discovered it, we can call it whatever we want. So we could call it, like, superkinetic energy or something. And that's basically what happens in physics when physicists get to a new concept.
They've got the job of describing it, which is why physics is sort of got so many weird words in it. But there's nothing special. It's just people naming stuff. There's nothing mysterious about it.
Differentiation is sort of the opposite to integration. Here you approximate the gradient with tiny little triangles at each point. So we would write that the gradient is a delta half mv squared divided by delta v. Again the approximation becomes perfect when these triangles are infinitely small.
And then you write out a differential equation which looks like this. The solution to this equation is mv which is basically back to where we were at the beginning of the video. So differentiation is a kind of inverse or opposite process to integration. That just gives you a flavour of calculus, but if you want to find out more, there's a really good playlist on calculus on the channel 3Blue1Brown, which you can get up here. I really recommend it, it goes into way more detail, and in fact the whole channel I love, it's so good.
So calculus is normally taught as kind of an abstract set of mathematical... rules, but to a physicist it's kind of the bread and butter of what we do day to day. You know, like a plumber has got a wrench, a physicist has got calculus.
When you learn physics in school you're given a bunch of equations and you have to memorise them and you have to apply them in different ways, but you're never really told where those equations come from, which is understandable, you don't have time when you're in school and it's kind of like a next level understanding thing, but when I was at university and I learnt calculus... and then I learned the derivation of all of those physics equations. It was a really satisfying thing to learn, because it was the first time where I really understood kind of why those equations existed, where they came from, and also how they related to each other.
And then when I started being able to apply calculus to come up with my own equations, that was incredibly cool, because it's like you're a freaking physics wizard or something. And so if you ever get the opportunity to learn calculus, I really recommend it, even if just for your own interest. It's definitely applicable. And if you're learning calculus and you don't really know what the point is, learning something that gives you the ability to derive new physics is a very good reason.
I should mention that algebra and calculus aren't the end to the mathematical tools that physicists use to derive new physics, but they definitely form a foundation which then you can build upon later or add on different things like matrix algebra. vectors and tensors. Depending on what path of physics you go into you'll need different kinds of mathematics.
If you're interested in learning more about derivations in physics and how to apply those to novel situations I recommend checking out the sponsor of this video brilliant.org It's a website where you actively solve problems that are broken down into chunks and you're encouraged to think carefully through each section and then bring them all together to solve the problem in the end. And the idea is to build up sort of intuition by seeing how the different parts are related to each other, that then you can apply that knowledge in novel situations more easily, rather than just memorizing equations where that's harder to do. So if that sounds interesting, check out brilliant.org.dos. Link is also in the description below.
And as an added bonus, the first 200 people to sign up can get a 20% discount off the annual premium membership, which unlocks all of their content. So check that out. And thanks for watching this video. Sorry about the delay from the last one.
I've been busy with Professor Astro Cat stuff. We just released a new book and I've also been writing the next book. But there'll be at least one more video this month.
I've got a whole load lined up.