 We're going to talk about what happens to things when they're undergoing constant acceleration now, and things undergo constant acceleration all the time around us, provided air resistance is negligible, because that's what gravity does. It took a long time to realize that air resistance was an important feature of falling things. Aristotle used to think that heavy things just naturally fell faster than light things, which looks like a reasonably good result of that experiment. But it's not necessarily the heavy thing and the light thing are acting differently under gravity there, because he didn't think about air resistance. Galileo put forward a sort of a thought experiment that made Aristotle's idea that heavy things fall faster than light things seem a little strange. He said, suppose you have one cannonball and another cannonball exactly like it. So because they're the same, they'll fall the same. Now, if we had a bigger cannonball instead, you'd expect that to fall faster according to Aristotle, because of gravity. I'm supposing you have those two cannonballs, and instead of having one twice as big, we just had those two attached by a little chain, like just a tiny little hair. That's now a bigger thing, right? So when I drop it, it should fall faster. But we can know that if we attach two cannonballs by a hair, that's not going to change how fast they fall at all. So that goes against our intuition. Oh, so is Aristotle? Okay, you have to make it thicker. They have to be properly attached, right? Well, how thick does it have to be? Does it have to be just a rigid little chain? What if I put a tiny blob of glue between them? Do they suddenly fall faster now? That doesn't really work with our intuition either. And in fact, what we understand now is that air resistance is really important in understanding why heavy things seem to fall faster. If I have my piece of paper, then it falls. It's falling because of gravity, but it's also being pushed up by the air. And the main reason that this pen falls faster than this piece of paper is because of the air resistance. And you can test that by taking the air away. And then when you've got no air resistance, you see that these two things will in fact fall at the same rate. Well, in my left hand I have a feather. In my right hand, a hammer. And I guess one of the reasons we got here today was because of a gentleman named Galileo a long time ago who made a rather significant discovery about falling objects in gravity fields. And we thought that where would be a better place to confirm his findings than on the moon. And so we thought we'd try it here for you. The feather happens to be appropriately a falcon feather for our falcon. And I'll drop the two of them here and hopefully they'll hit the ground at the same time. How about that? So that's got to be absolutely the world's most expensive version of that experiment ever. And yet it wasn't even in HD. You can get the same effect just by putting things in a vacuum tube and sucking out the air. Now Galileo didn't just think about gravity and said, I don't think Alice Dottel can be right. He actually went and did the experiment. And when he was doing the experiment he didn't have really great clocks and things fall pretty fast so it's hard to get accurate time measurements. So he slowed things down by putting things on a slope and rolling them down the slope. But he tried different slopes and he found that indeed gravity caused things to accelerate at a constant rate. So to keep things simple we'll just start by considering motion in a single direction. So we have some object that we can define and it's traveling in some direction. It's really important in your diagrams to be as clear as you possibly can. And so we're going to have a before diagram and an after diagram. And you'll note I've explicitly said after a time t for my diagram so I've kind of defined my symbols in my diagram and that's good practice. So after a time t it's going to be somewhere else. Now in order to figure out how it's going from one to the other we need to know its acceleration and velocity just because it started on the left and so it started with a negative position doesn't mean it was going to the left. It could have had an initial velocity where it was heading to the right. So let's look at the initial velocity so the initial velocity is a factor it's got a direction so it's either positive or negative if we're going at one dimension. And we're going to call that u so that's its initial velocity. And we're going to have to have a final velocity and the final velocity can go in either direction so that's positive for the purposes of this diagram. And of course the reason we're working all of this out is that we've got an acceleration so after a time t of acceleration and again I'll direct that to the right assuming the acceleration is in the positive x direction. Let's look at what happens graphically before we worry about the algebra and so you have the acceleration doing nothing at all over time so if I have time on one axis and acceleration on the other we've already drawn on our diagram that it starts at a and it does not change that's our idea we're talking about constant acceleration so it's like that we've drawn it above the origin because our diagram has a positive number pointing to the right. Now what happens to velocity? Well we know that the velocity is changing at a constant rate so it starts at u at time t equals zero and then it's going to increase because our acceleration is in the positive direction it's going to increase up until it gets to our final value of v and that should be a lovely straight line I've drawn a reasonably straight line now what's going to happen to our position if that's going on? Well we start at a negative value on our diagram so I'll have that there so that will be our x naught now if I'm going to write x naught on there I should write an x naught on here always have things defined on your diagram and if we're going to be moving and we end up at x final at a later time now what's this shape going to look like? Well it's going to start off with some slope and the slope here is going to be the slope given by our initial velocity which is u so if we break this up into a small amount of time delta t then we know that the velocity is the distance travelled over the time and so the distance travelled is going to be the velocity times the time or u times delta t and up at the end it'll work the same except for delta t and somehow those slopes have to match and so if we try and smoothly connect those curves it's got to look something like that let's do the algebra for that so we know that the acceleration is constant and we know the definition of acceleration so it's the change in velocity divided by the change in time and so the change in velocity is going to be the final velocity minus the initial velocity and the time taken we've written down as just t if we multiply both sides by t if we multiply the left by t and the right by t it's going to cancel that and so we'll end up with at equals v minus u and we can rearrange that the most common way of arranging that one is as v equals u plus at just adding u to both sides so that's the first equation we often remember when we're talking about constant acceleration that relates the final velocity to the initial velocity and how much you're accelerating it's a little bit harder to figure out where we're going to be because our velocity keeps changing so let's just look at a particular time in the middle where we've got say a velocity v1 at that time if we look at a very small amount of time the velocity won't change very much and so if we take a small amount of time delta t then we know that the distance it's going to change is going to be delta x is going to be v1 times delta t and v1 times delta t is basically just a little rectangle that big it's the area of that rectangle and what we have to do is we have to have the little change in distance we get from that piece and the little change in distance we get from that piece and so on and so on and that turns out to be just the area under this curve that's how far we go and again you can do that formally by going to calculus but if you're just believing that that's the area and you can kind of see it it's just the sum of all those little rectangles then it's the area of this rectangle plus the area of this triangle and the area of this rectangle well that height is u and this length is t so that's just u times t and this height is v minus u and that length is t so this is going to be half of the product of those so it's v minus u times t that's the size of that big rectangle there and because it's the triangle we're going to have half of that so we're going to have to add this area together to get the distance that we travel so the distance we travel is the final position minus the initial position and we've just agreed that it's going to be this ut the area of the rectangle plus v minus ut on 2 which is the area of the triangle alright so I'm going to introduce a symbol which everyone else uses which is just s the total distance traveled and then I'm going to plug this velocity here into there because it was particularly easy because we already had v minus u so we could just go straight in and plug in our at and that's the second equation we use to describe how things move under constant acceleration and so we can see this curve here is quadratic because it goes as time squared and there's a third equation that people use which is really just a rearrangement of these two what we're going to do is we're going to try and eliminate time and so from this equation here if we divide by acceleration instead of by time we get and then we're going to plug that into here so we just expand out the first bracket and then the second bracket and then we note that that term cancels with that term and we note that this term half cancels with that term and we got two things left and if we model by both sides by 2a and that's our final equation relating the initial and final velocities and how far you travel for constant acceleration