In physics the equation for momentum is this P = mv. So if you have got a certain mass,
moving at a certain velocity, it has momentum equal to the mass times the velocity.
Like any equation you can draw this as a graph. I have chosen the axes to be the momentum
and velocity, and the mass gives the gradient or slope of the line.
Now lets 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 in for P, because P = mv. So that gives the
area to be a half times m times v times v, or a half m v squared.
Now you may be familiar with this equation already, it is 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 = mv in the example beforehand.
Calculus involves two parts, in integration and differentiation. Integration is where
you are finding areas underneath curves, and differentiation is where you are finding 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 and so calculus is an
incredibly useful mathematical tool.
Now we didn’t actually do calculus in my example because the graph is a straight line
which basically means it is easy enough to just use simple geometry to find the area,
and calculating the gradient of a straight line is simple, it is 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 a curved line, like the plot of kinetic energy
that we’ve derived. It’s a curved line because it has 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 teach you all of calculus in this video, but I’ll give a brief description
the way it works. The idea behind integration is that you start by approximating the area
under your plot with a series of rectangles, so we can write that each area is a half m
v 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 gets more and more precise, and becomes perfect when the
width of the rectangles gets infinitely small. Then we re-write 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 us physicists we don’t have a name that describes the quantity that
we have just found, so we are already sort of in the wilderness of physics. A physicists
description of it it would be the integral of kinetic energy with respect to velocity,
but seeing as we have discovered it we could call it whatever we want so we could call
it, like, super kinetic energy, or something. And that is 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 is 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 a delta
half m v 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 is
a really good playlist on calculus on the channel 3Blue1Brown which you can get up here,
um, I really recommend it, it goes into way more detail, in fact the whole channel I love,
it’s so good.
So calculus is normally taught as an abstract set of mathematical rules, but to a physicist
it is 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.
And when you learn physics in school you are given a bunch of equations and you have to
memorise them and you have to apply them in different ways, but you are never really told
where those equations come from. Which is understandable you don’t have time when
you are in school and it’s kind of like a next level understanding thing, but when
I was at University and I learned 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 er cool because it’s like you’re
a frikking 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, um, it is definitely applicable, and if you are learning calculus
and you don’t really know what the point is, learning something that gives you the
ability to derive new physics is, is a very good reason.
I should mention that algebra and calculus aren’t the end of the mathematical tools
that physicists use to derive new physics, but they definitely form a foundation which
you can then 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 interesting in learning more about derivations in physics and how to apply those
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content, so check that out. And thanks for watching this video, sorry about the delay
from the last one, um, I’ve been busy Professor Astro Cat stuff, we’ve just released a new
book and I’ve been writing the next book. But there will be at least one more video
this month, I’ve got a whole lot more lined up.