The Origins Podcast with Lawrence Krauss - Physics for Everyone, Lecture 3: Motion, from Galileo to Dark Mysteries

Episode Date: April 14, 2026

We usually begin the study of physics with a discussion of motion, not because it is easy, or because the modern understanding of motion began with Galileo hundreds of years ago. Rather, Galileo’s groundbreaking work provides a paradigm to understand how physics is done today. Extracting out the fundamental essence of motion from all the distractions associated with what turn out to be irrelevant complexities was a monumental intellectual leap for humankind—a leap we often take for granted. Without the leap, for example, Newton could never have made his profound discoveries about the relationship between force and movement, nor his discovery of the Universal Law of Gravitation. But too often we treat these remarkable achievements as something belonging in antiquity.. as if we have moved far beyond them in every way. Nothing could be further from the truth. Applying the very same ideas that Galileo and Newton developed leads us to the cusp of modern physics: the discovery of the dominant mass in the Universe, a vast invisible sea of dark matter. In this episode, we travel over 450 years of physics, from Galileo, to the threshold of our understanding of the cosmos today. Hang onto your hats. I’m also pleased to share a quick PSA. A reminder of our 2026 Origins expedition through the Greek archipelago (July 24 to 31), with a Cyprus add-on (July 17 to 22). If you’re interested, it’s worth raising your hand early. These trips tend to fill quickly. Express interest at https://originsproject.org/greek-adventure-2026-application/As always, an ad-free video version of this podcast is also available to paid Critical Mass subscribers. Your subscriptions support the non-profit Origins Project Foundation, which produces the podcast. The audio version is available free on the Critical Mass site and on all podcast sites, and the video version will also be available on the Origins Project YouTube. Get full access to Critical Mass at lawrencekrauss.substack.com/subscribe

Hi, welcome back to Physics for Everyone, Lecture 3, in trying to understand the way physicists understand the world from the simplest physics to the forefront of our understanding of nature. In the last lecture, I talked about several things from approximation and order of magnitude estimation and dimensional analysis, the fundamental tools that affect physicist's worldview to take a complicated universe and turn it into one we can understand. Now I want to start talking about how and what we actually understand about the universe and I want to do it by jumping in

To what is probably the most complicated thing in all of nature and that complicated thing is nothing other than motion We often start physics classes talking about motion for a reason But that tends to suggest to people that motion is simple and we're gonna get to the good stuff later Well you'll see that in fact the ideas that Galileo and issue and others to understand motion became a prototype for understanding the rest of the universe and by the end of this lecture

actually we'll get to the forefront of modern physics, I'm hoping. So to give a sense of how non-intuitive motion is, I just want to give a pop quiz, the kind of thing I like to begin physics classes with every now and then. So this is a pop quiz.

Draw on a piece of paper this trajectory of this particle. Okay? And then this is the instance. when I want you to think about what happens next. So the particles are moving it, and then I'm not going to do anything to it. I'm going to leave it on its own.

What happens next? Draw its trajectory after that, as effectively as you can. So take a few seconds to do that if you wouldn't mind. Okay? Hopefully now you've drawn the trajectory, and what is the actual trajectory? Well, some of you may remember

from high school physics, others may not, that the actual trajectory is a straight line. Independent of what was happening before, if I leave it on its own, if I leave a particle on its own to travel and do not disturb it at all, then the particle will continue on a straight line

with the speed that it had at that instant. Okay? And that was what Galileo realized and defied centuries, actually millennia, of thinking about motion, and he did it by thinking about the world in a new way. The fact that it took Galileo to redefine our thinking about motion, and intelligent people from Aristotle and onward

had thought about how things moved in the world and gotten it totally wrong, suggests that motion is not so simple. And how Galileo dealt with it really is a prototype for how we deal with the rest of the world in all of future physics. And really the simple idea is that sort of stand outside the box. He began to think about what we see and what we see may be, we may be losing the forest for the trees.

We may have to step back and ask, well, is it really this way or is it just an accident of us looking at a complicated world? Try something, seeing if it worked, as we'll see, and then running with it as far as he can, not literally, but intellectually. And so how did you do it? Well, this is a, I hate to you a sport metaphor is actually, because I'm not a big fan of sports, but this is a football field, an American football field. And the thing that Galileo realized was that position is unimportant.

By the time, for millennia, people had focused on the position of a particle. And Aristotle had basically said, you know, all objects seek to basically go at rest, to the lowest point of their trajectory and stay there. Somehow that was a special place. All objects seek the ground in some ways and then stay there, or all objects get to a point and stop. And Gallia realized the position was unimportant. And it's a kind of thing in football, which is kind of interesting

because when people are talking about football, when people are near the end zone, somehow the position seems important. But of course, it isn't. The dynamics of what's the, happening at the 10-yard line is the same as the 50-yard line. What's really important is not where you are, but how you're moving along the field. And that's the jump that Galileo made to go from thinking about position, which is really important because every point on this

football field is effectively the same as every other point in terms of what you're going to do with plays and the laws of physics, the government. But what really matters is position, is, sorry, is motion. Now, when thinking about this, motion, as I say, is complicated because we see where things are. But sometimes we can't really see the motion because we're seeing time-lapse versions

of where things go. And when an object is moving, it's moving at some speed or velocity. And I don't want to get hung up on these ideas. In physics classes, we spent a lot of time talking about speed versus velocity, velocity is speed plus direction, it's a vector, but we're not gonna worry about that stuff here.

Objects are moving with some velocity V, and when I say velocity, I'm basically interchanging with speed here. And to understand what happens to their speed, the first thing, Gallo realized, is when an object is at rest, and now I'm gonna move to the whiteboard here, when an object is at rest,

its speed is zero. Well, that's not very profound. But he also realized, okay, when it's at rest, it stays at rest. So its speed doesn't change if you don't do anything to it. And he started to think about this and said, well, V equal zero, speed equals zero, is just one example, one value for the speed of an object. If it's not zero, okay, could it be that objects continue to travel at the same speed? Now, as I say, Aristotle and others had for a long time argued that this was not the case,

because, of course, things fall due to gravity we now know, and things slow down when you push them on the chair. But what he did was, once again, trying to abstract out the fundamentals. The cow is a sphere, if you wish, from the complicated shape of this cow. And he started to say, well, yeah, it's true, objects slow down, but they slow down at different rates. On this rug here, they slow down faster than if they were on the floor over there. And on the ice, we all know

if you continue to roll or in a marble floor, you'll continue to slide faster. So he suggested that perhaps the slowing down was extraneous. The fundamental facet of motion was that objects would continue to move unless you did something to them. And the reason they slow down is that the floor or the rug

are doing something to them. And the argument is quite simple. And once again, if an object is at rest, it stays at rest, but at rest is just one value of velocity. There are many, many values that can have a velocity.

And he said, therefore, if the velocity isn't constant, isn't zero, it still remains constant. Sorry, I shouldn't write, it isn't constant. If it's not equal to zero, then it remains constant again.

And that, single realization changed everything because it set the stage for thinking not just about constant motion, but as we'll see, motion that's changing. But even constant motion tells us something interesting. If an object is moving and in general, unless you do something to it, remains moving at the same speed, that led Galileo to what we now call Galilean relativity, which is that you cannot tell tell if you're moving or standing still, that the laws of physics that you measure in a laboratory

that's moving at a constant speed will be the same as the laws of physics if you're standing still. Now that's in principle obvious to those of us who say have taken, if you're on an airplane and you forget the noise of the airplane, if you throw a ball up, it'll come down back into your lap. And if the windows are closed, other than the noise and the bumps, you might not be able to tell you're an airplane or a train, particularly in Europe, where they're smoother,

again, you won't be able to notice if you're moving, you're standing still. In fact, many of you may have had this experience in a subway train or in a train station where you're sitting in a train and you look at a train in the neighboring tracks and you start to move relative to it. And for a moment, as long as it's not bumpy, you don't know whether you're moving or whether the other train is moving. And at constant speed, you cannot tell you. tell whether you're moving. There's no experiment you can do in the laboratory other than looking outside

that tells you you're moving if you're moving at a constant speed, nothing is different than if you're moving at zero because a constant speed is just, a non-zero constant speed is one example of uniform motion. Uniform motion, as he defined it. Now this may sound, as I say, trivial in retrospect,

although with the example I gave you, you can see that it may not be so trivial, that people realize that things, most people think that somehow the future motion of an object is determined by the past motion. It isn't. It's determined by whether you're doing something to it at that instant or whether it's moving freely. But the really important thing of course about that is that it sets the stage for realizing that if you don't do anything to an object and it continues to move at a constant speed, then if you want to change, the speed of an object, you have to do something to it.

Again, it may sound obvious, and in retrospect of course it is, but it took, eventually you can see why this set the stage for Newton. So if you don't do anything to something, then the velocity in object is constant. If you do something, then the velocity isn't constant. Now, how do we define a non-constant velocity? velocity. The velocity is changing and it's changing with respect to time. At one instant you're going five miles an hour, the next instant you're going to ten miles an hour. So we define the acceleration of an object as the rate of change

of velocity with time. This is a particular, and I can write it in mathematics, it's the rate of change of velocity over time, but that doesn't matter, but that's how we would write it in a simple form. But acceleration is particularly non-intuitive, because we never see acceleration. We feel it, but we can even see objects moving. But to determine the acceleration of an object, whether it's speeding up or slowing down,

is much harder to see in general. And in fact, one can do whole physics classes to talk about how non-intuitive acceleration is. But you see, the point is that if this is, So this is true if you don't do anything. This is the case if you do something. Now, what do you do? You act on the object.

And of course, Newton realized, and this is formalized in obviously his most famous equation, perhaps, that therefore I can define a force to be that which changes the motion of objects. And so no force means velocity is constant, a force not equal to zero means the velocity is not equal to constant, you get acceleration. So the force is proportionally acceleration or another way of thinking about it, the acceleration is proportional to the force.

No force, no acceleration, okay? A force acceleration. Would you ever have something like that with the proportionality, the constant of the proportionality, in a sense, he sort of defined as mass. It's harder to move a heavy object than a lot. light object and therefore the same force applied to a heavy object will produce less acceleration than it will to a light object. If this is some number and it's equal to the product of these two numbers, if M is very

big then A will be very much smaller in order to produce the same number here compared to a light object. So it's a lot easier to move a light object than a heavy object and we can call this, you know, people call it the inertia and the and and and, and, and, and, and, and, and, and, and, and, and, and, and, inertia and mass, it turns out they have, depending on how you measure it, they can be different things. But that's, again, something subtle that I don't want to go into here. I want to, we all have an intuitive idea of this.

And of course, this is Newton's famous law. And if you want, you can think of this as defining. Well, acceleration we can measure directly. We can see whether something's moving five meters per second, one second, and ten meters per second, another second later. And if you wish, depending upon what you want to take is given, if you think you can measure the mass, and you know it's a one kilogram object,

this defines what force is. Alternatively, if you could somehow measure the force by the tension on a spring, then measuring the acceleration the object will determine the mass of the object. So it really defines mass or force. And these are, these are, these are, in principle actually subtle things that one needs to talk about in a physics class,

but I'm not gonna go, they're not necessary here. The only thing I wanna point out is that what is therefore the simplest kind of non-uniform motion? The simplest kind of uniform motion is when velocity is constant. The simplest kind of non-uniform motion is therefore, and you see, until you focus on velocity, it's hard to think about what non-uniform motion is.

The minute you focus on velocity and say, well, velocity is constant is the simplest kind of uniform motion. The simplest kind of non-uniform motion is clearly where the acceleration is constant. The rate of change of velocity is constant. That's the simplest kind of non-uniform motion, and it was the kind of non-uniform motion

that Galileo first looked at. You can see, in a sense, as Newton once said, he got where he did by standing on the shoulders of giants, He was actually making a derogatory comment about some competitors who were much shorter than him. And he wasn't a very pleasant fellow, Newton. But you really see, in a sense, he did stand on the shoulders of joints.

Because without the framework of understanding motion, velocity, non-uniform motion acceleration of Galileo, in some sense, it wouldn't have been so obvious for Newton to come up with the F equals M.A. and you'll see the other things that he did. So it was Galileo first studied non-uniform motion because he said, well, that's the most simple,

and the kind of non-uniform motion was constant acceleration. And he showed that the simplest kind of non-uniform motion, constant acceleration, was exactly what objects experienced when they fall. That was, again, profound. And it was felt for millennia that when you let an object go, it immediately achieves its final velocity instantaneously. And it's kind of amazing, once again, how people accepted that wisdom without thinking it through.

Galais was famous for thinking of these thought experiments. And he said it's obvious to anyone, if you really think about it, that objects are speeding up. And one way to think about it is hold a bowling ball one inch above your foot and let it go. and then hold it three feet above your foot, you don't want to let it go. It's clearly hitting you much faster.

Although, because the acceleration due to the earth is large enough, for our eyes, it looks like things immediately sort of achieve their final velocity. But what he noticed were things that were uniformed, he did some experiments, which again is the prototype of modern physics. And he said, well, and he discovered something rather interesting. When he measured the position of an object that was falling, say on an inclined plane,

he noticed something really amazing, that if you looked at the position of time two minus the position of time one, you got some number. And then position of time three minus the position of time two, you get another number. he noticed that something really rather was rather remarkable. That if, that, that these things were in proportion to odd numbers. If this was one, this would be three. The next one would be five. The next one would be seven.

So X four minus X three. The rate, if this was one unit, this difference would be three units, this difference would be five units, and so on. And the fact that it was always. just these progressionals, one, three, five, seven, nine, whatever, was what he recognized was the observable characteristic that he discerned from uniform motion. We now actually understand where this comes from, and we can understand it a few different ways

actually using the kind of tools that I taught you in the last lecture. You know, certainly one thing that should be clear is if the velocity is constant, then the distance an object traverses is the velocity times the time. If I'm going 30 miles per hour, in one hour I will travel 30 miles. In two hours, I'll travel 60 miles.

So the distance is proportional to time if the velocity is constant. One can show, and we won't bother showing it here, that in fact, if the velocity, if the acceleration is constant, then the distance traveled by an object is not proportional to time, it's proportional to time squared. And this will produce this because in fact if let's just look at this in one second the distance if the if the constant proportionality is one in one second just the distance travel will be one unit one meter one mile I don't care

in the second one two seconds will be four this third third after three seconds will be nine so one four nine well four minus one is three 9 minus 4 is 5, and then the next, after 4 seconds will be 16, 16 minus 9 is 7, and you can see that this is equivalent to this, that the distance is proportional to the square of time. The actual constant proportionality is 1 half the acceleration times time, D equals, let me write it down, D equals 1 half A, T squared. Doesn't matter, I don't care about the equations here particularly,

But I want to show you, by the way, that using the tools of dimensional analysis, we could absolutely have shown that the distance is proportional to time squared without having actually derived it if for a constant acceleration. Why? Because this would say, clearly the distance is going to be proportional to the acceleration. The distance traveled is going to be larger if the acceleration. is larger, et cetera. Now, what are the units of acceleration?

It's rate of change of velocity over time. So velocity has units of, say, meters per second. So the units of acceleration are meters per second per second. How many meters per second do you change every second? So an acceleration of five meters per second says after one second you're going five meters per second. sorry, five meters per second squared is the acceleration. After one second, you're going five meters per second, after two seconds, you're going ten,

after three seconds, you're going fifteen, et cetera. So this has units, this has units of meters per second squared or distance over time squared. And therefore, if you want to say what distance is equal to, and one thing, one of the terms in the equation has units of distance over time squared, then you have to multiply by time squared to get distance in the end.

So once again, dimensional analysis will tell us that, you know, it won't tell us the constant of proportionality that there's a one-half here, but it'll tell us when I have A being constant, that T, that the distance goes as a square of time. And, in fact, what we now know is that the acceleration due to gravity, which is constant at the air surface,

which is what Galileo measured, is 10 meters per second squared. And that means after one second, ignoring air resistance and et cetera, for a bowling ball falling or a cannon ball falling from the Tower of Pisa, after one second, it's going 10 meters per second, after two seconds, it's going 20 meters per second.

I actually did an experiment the other day to test this. And I fell in this pit five days ago. In the dark, when I was walking, it's about a seven foot deep pit. And I've survived, which is fortunate, for me at least, and unexpected. And you may notice the gash in my head. There's many more gashes in the rest of my body, happily, other than being in extreme pain and bleeding, bleeding, I survived this, but I have written on my bandage here, A equals 10 meters per second

squared, to remind me that I shouldn't fall into pits at the very least. Now, that's the heart of motion. Constant velocity, constant acceleration, and constant acceleration is relevant because it's not only the simplest form of non-uniform motion, but also the most of motion. most ubiquitous in the sense that at the earth's surface, objects fall at a constant speed, a constant rate of acceleration, not a constant speed, sorry. And of course, that's the substance of Galilei's famous experiment in the leading tower of Pisa, showing that, in fact, all objects independent of their mass fall at the same speed. Once again, it's very non-intuitive.

I once was talking to the leaders of the free world at a lecture, and I held out a book and a piece of paper and said, if I let them go, which will fall, which will hit the ground first? And everyone said the book, I let them go, the book fell, and the piece of paper fluttered down. And I said, why?

An amazing fraction of these significant political leaders said because the book is heavier. And then I did the kind of experiment that Galileo would do, which is, I'm going to come over here for a second, off screen. And, you know, when I dropped the book, it fell,

much more slowly, but then I took the same piece of paper and scrunched it up and let it fall, and then it fell at the same speed as the same rate of acceleration as the book, and clearly it wasn't any heavier. But nevertheless, once again, it took Galileo to recognize that, and in order to recognize that, he first had to understand velocity, non-constant velocity, and constant acceleration. And that's the source of his famous experiment in Tauropisa. But he did something else, something else which is essential for Newton and will be essential in some sense to lead us to the discoveries of the 21st century. And that is one of the wonderful things about physics is if something works, we tend to copy it. And you'll see that over and over again, including in

this lecture. And we copy it until it doesn't work anymore. the ideas, and so we just check to see if something works. And one of the famous experiments that I used to like to do when I lectured physics, large undergraduate physics classes, is ask the following experiment, well, let me just put it this way. Take LeBron James basketball player. God, I'm using all these sports analogies, I can't help it. But in this case, so if LeBron James wants to jump,

as high as he can to make a basket. Does he need to run as fast as he can to do that? And most people say yes. But in fact, LeBron James' vertical motion is independent of his horizontal motion. This is what Galileo first said, that two-dimensional motion is just two copies of one-dimensional motion.

So if an object is moving up and this is the X-direction, and this is the Y direction, then the motion in the Y direction is independent of the motion in the X direction. And in fact, this is important because if we look at objects and if this is the surface of the Earth

and this is vertical and this is horizontal, when an object moves, in this direction, we know gravity is always pulling down. So there's a force pulling down towards the Earth, but there's no force pushing,

you know, again, ignoring friction in this direction or this direction. So let's think of what the trajectory of an object is in two dimensions. Well, if an object's moving the X direction, its distance in that direction is proportional to time. But if it's falling at the same time, the distance it falls is proportional time squared. Now, the first thing is, because these two are independent,

it means that how far you fall in a given time is independent of how fast you're moving. If I asked most of you, perhaps before I drew it this way, if I took a, in fact, I could have done it with two coins, and again, a demonstration we do often undergraduate physics labs, if I took a bullet and a penny and I shot the gun at the same time as I dropped the penny, if I asked you which would hit the ground first, the penny or the bullet, and you can ask all your friends this, almost everyone will say the penny will hit the ground first. The penny will hit the ground first. The penny or the bullet. and you can ask all your friends this, almost everyone will say the penny will hit the ground before the bullet. Because somehow the bullet is moving so fast that you don't see it falling. So it feels like it's just heading horizontally almost forever, okay, without falling. But that's just because it's moving so fast you can't,

that it goes out of the frame in a time scale so short that even the penny is not fallen very fast. But because these are independent, they will both fall at the same rate, but they'll do it with different trajectories. If I take a penny, it'll just fall. But if I take the bullet, because it's trajectory in the X direction is proportional to time, whereas its trajectory in the Y direction is proportional time squared, it will look like that.

That means because this goes like time squared and this goes like time, this defines something we call a parabola. and the motion of objects that are moving at the earth's surface that are moving along and let go due to gravity is always parabola. The faster those objects are going, the more stretch out is the parabola. And this is, you know, so a bullet, when we see it in the room, looks like it's not falling at all, but eventually it'll fall.

And in the same time, it'll hit the ground here as the penny will, but by the time it hits the ground, it'll be way out of sight. Okay, great. Now the reason I drew this is because one of my favorite drawings in all of physics comes from Newton's Principia Newton began to think of something more interesting. He said okay, well this is true. This was all Galileo But Newton said okay, well this is interesting. So I have a cannonball But it's on the surface of the earth and the earth is a sphere So let's say from a mountain I shoot off a A cannonball. Well, it's gonna it's gonna do it

the trajectory is going to be a parabola. And if I have a faster cannonball, it's going to go farther. But at some point, the fact that the earth is curved, if it's fast enough, will affect where this hits. Of course, if it's nearby, the earth seems flat.

But when you're sending things fast enough so that they don't hit the ground until they've covered a significant fraction of the earth's surface, they begin to sense, if you wish, the curvature of the earth. And then he asked, What happens if I shoot a cannonball fast enough, so it keeps falling and never quite hits the earth?

He said, maybe it'll come back to where it begins. This, we call an orbit. And this was a crucial way of thinking because it demonstrated something that was up until that point totally misunderstood. Objects, it looked at that time as if all objects were orbiting around the Earth, the sun, the planets, etc., etc., etc. But the idea was how do you get put something in an orbit? How do you get something to go around the earth? And the thought was the way you do it is you push it. You keep pushing it around But you can see once the cannonball has let go here nothing's pushing it. It's moving at a constant speed All that's happening and this was the key point was that it's not being pushed in the direction of its motion

it's being pulled always downward towards the earth. And this was the crucial realization that led Newton to his greatest discovery, the universal law of gravity. Now, how could he determine the nature of the gravitational force? Remember, acceleration is the rate of change of velocity with respect to time.

Now, this is moving, this cannonball is moving at a constant speed. But remember I said velocity is speed and direction. So at time one, the velocity is in this direction, at some time T2, the velocity is in this direction. So V1 and V2. Now the magnitude of the velocities is the same. The object is not speeding up or slowing down, but its direction is different. And therefore, because the direction is changing, the velocity of an object is changing, and therefore it's accelerating.

And he said, let me think about what the acceleration looks like. Well, let me just draw velocity vector at one time and the velocity vector another, the same magnitude. And the difference between them is what we call the rate of change of velocity. This thing here. Now, I'm going to go through this, and if the math seems a little daunting, it's not. really very daunting, but if it does, just accept the result. The difference between this and this object is an angle theta. And let's think about what, so the object is moving around from here to here in some time. So let me, let me draw it in a way which looks a

little more simple. I've made the diagram look a little complicated. Okay. So it's gone from here to here in that time. What distance says it's gone? Let me draw the... So it traversed some angle, which I'll call theta, and the distance has traveled, since it's moving at speed V, is VT. And if it's moving in a circle of radius R,

then the angle, for small angles at least, the angle is VT over R. Okay? The angle that it traverses in a circle you can show by trigonometry is just the size of this over the radius of the radius of the sphere. So the longer you take, the angle is bigger and it grows with time, and I should make big R because that was a big R there. Okay. Now, if I think about, so that's what this angle is here, but clearly if the velocity is bigger, if the velocity is bigger, then this delta V grows.

And so delta V is basically proportional to the velocity itself times the angle that the object has traversed. And therefore in a time delta T, so at a time delta T, the angle traversed is V times delta T over R. and this therefore becomes v times v over r times delta t and therefore delta v over delta t is v squared over r this was Newton's realization that the that two things that when an object is going in a circle it's accelerating the direction of the acceleration if you look at it is always as always point towards mainly this line here, this delta V,

if I look at two slightly different times, the delta V will always be pointing towards the center of the Earth or the center of the object. So the acceleration is always pointing inward towards the center and its magnitude is v squared over R. So, again, you may say, big deal, why do I care about this? Well, this changed our picture of the world. Newton showed,

that to go on a circle, an object is accelerating, and the magnitude of the acceleration is v squared over R if the object is moving at a velocity V, and the acceleration vector is always pointing towards the center. Now, this is important to him, because remember, force equals mass times acceleration. So what Newton said is, to make something go in a circle,

you apply force equal to the mass of the object times the square of its velocity over R, and you apply it towards the center. We all know what's pulling things towards the center. It's gravity. So the magnitude of the gravitational force is mv squared over R if the object is moving in an orbit R. Now, once again, Newton stood on the shoulders of giants.

Because a generation before him, Johannes Kepler had discovered something remarkable. Kepler was an interesting guy. He was assistant of a really eccentric and ridiculous Danish nobleman, Tico Brahe. Tico Brahe was an astronomer at a time when there were no telescopes, but he carefully observed the sky at night. He saw an exploding star.

that was visible, but he was the first person discovered. The Emperor of Denmark was so enamored with this that he gave Tycho Brahe, Tico Brahe, his own island, and he was a feudal lord for a while, an awful feudal lord. Kepler was assistant, his assistant, what Brahe did remarkably and carefully, in an observatory I've been to him, what is now the island of Venn, which is now part of Sweden, but once was part of Denmark, if you go there, his observatory

he's underground. Why is it underground? Because in fact he didn't use telescopes, he just used quadrants and careful instruments, and it was in a hollowed out region so the wind wouldn't blow there, and you could very carefully measure the position of the planets on the sky. And what Rahe did after, it was just measure those positions of the planets on the sky and measure it better than anyone in all of history had measured it, better by a fact of factor of at least 10 than anyone had measured before. And then he gave Kepler the data, basically, and said, figure this out. Kepler, by this point, Brahe had been exiled again because he was a crummy feudal lord.

And Kepler went to Warsaw, if I'm right. In any case, Kepler was, unlike Brahe was not a, was not a nobleman, his whole life he'd experienced problems. He was fascinating guy. He wrote basically the first science fiction. story of going to the moon. His mother was arrested and prosecuted as a witch, and he defended her successfully. So his life was one trouble after another, but he spent almost 20 years analyzing that data and discovering several remarkable things about the orbit of the planets

in the sky, about their positions and their orbits of the planets in the sky. Basically, He discovered, of course, that the planets were not moving around the Earth. They were moving around the sun. And his three laws of planetary motion are fascinating. But the one that really matters is he discovered that Kepler's data, and here's the actual data of the motion of the planets moving around the sun, their speed versus their distance from the sun. This is data taken today.

This is the fundamental law that one of Kepler's fundamental laws it's usually stated differently in textbooks, but Kepler discovered that the velocity squared of planets around the sun is proportional to one over the distance from the sun. Now, this gave the crucial information that Newton was able to have that allowed him to discover the law of gravity. Because remember, now it turns out Kepler also showed

that the orbits of the planets around the sun wasn't quite circles, was actually ellipses. But again, that, for our purposes, there are approximately circles, or at least, let's say the Earth's orbit on the sun has an eccentricity. It's very small, 3% or so.

So to first approximation, we can think of it as a circle. What's crucial is that Newton had already shown that to make something go in a circle, it must have a force mv squared over R. But Kepler tells us that v squared goes like 1 over R. So if F is equal to mv squared, squared over r but v squared goes like one over r this means the force is proportional to

m over r squared but of course it's clear that heavier objects probably probably a greater force and therefore there should be in the constant proportionality should be the mass of the object that's exerting the force that's one way of thinking about it the other way we normally describe it is that that if the if I'm pulling you with a force that you feel the equal and opposite force pulling me, that was one of Newton's laws, and therefore if it's going to be reciprocal, if the Earth is attracting, if the Sun is attracting the Earth, the Earth is also attracting the Sun.

So the force is proportional to the product of the masses in 1 over R squared, and therefore when there's a proportionality, we can give it a constant, and we call it Newton's constant now, but this is Newton's famous law of gravity. was able to derive the universal law of gravity by using the results of Kepler and the crucial fact that he discovered what you need, what the magnitude of a force must be to produce a constant acceleration of associate with an orbit.

And that led him to this fundamental discovery that changed our understanding of nature. The first universal law of nature that some people say, in fact, ended the burning of witches. Because once you could show that even the motion the planets around in the sky was predictable and explainable by natural effects, namely the force of gravity, then maybe things like natural disasters were also due to natural effects and not witches and things like that.

Now great, but we can use this, universal law of nature, we can use this to discover something. And the last thing I want to talk about in this lecture is how we can use it. today to discover something fundamental about the universe, which was such a huge surprise that I know it's going to, at least win one Nobel Prize, probably within this century. But before I get there, I said it was a universal law of gravity. Well, it applies to planets. But how do we know it's universal?

How do we know the same force of gravity that's causing the planets to go around the sun is the same kind of force as we experience on earth. Well, of course, there's the famous apocopal story of Newton sitting under an apple tree. Looking at an apple falls, an apple tree, and an apple falls. And Newton is already, actually, a Galileo, making a smiling apple, Galileo's already told us the apple falls with an acceleration of 10 meters per second squared. But now we can say,

well, there's something else that's falling around the earth and that's the moon up in the sky. Because remember, the moon is going in an orbit around the earth and what Newton had shown is that an orbit is nothing other than continually falling. And the moon is attracted, if it's universal, by the same gravitational force as the apple on this surface the earth.

And if he could show that, then he could prove that it was universal. And you can, because we know the acceleration at the surface of earth is 10 meters per second squared, but because the force of gravity is proportional one over the distance of the object from the center of the earth, the earth is about 6,000 kilometers in radius,

or as we like to say from the other day, six times 10 to the third kilometers. The moon out here is about 400,000 kilometers from the earth. So the moon, distance from the moon to the earth, is four times 10 to the fifth kilometers. And if the forest goes like, one over the square of the distance from the earth, then the acceleration of the moon,

if we know the acceleration of this, of an apple right at the surface of the earth, the ratio of the acceleration of the moon towards the earth compared to the apple of the earth towards the earth should be the ratio of the square of the ratio of the distances for the center of the earth. So if the moon, if this is four times 10 to the fifth kilometers and this is six times 10

to the three kilometers, kilometers, this ratio is roughly 70, and therefore the ratio, if that's the ratio of their distances, and if the force of gravity goes like one over the square of their distance, and if the acceleration due to gravity is proportional to the force, then the ratio of their accelerations should be 1 over 70 squared, or about 4,900. So the acceleration of the moon, compared to the acceleration of the, of the, of, of, of, of, of, the apple should be the ratio of those one over the ratios those accelerations show and the if the if the apple is going at 10 meters per second squared in the earth and and

the the the acceleration is now smaller by 4,900 is this should be over 4,900 and that should be the ratio of their accelerations. Now, I think because I don't want to bother with the, the, I can plug in the numbers now and say if this is their acceleration and if acceleration is v squared over R, then that tells me what the velocity of the moon is around the Earth compared to, if I start at 10 meters per second squared

and divide, I get the acceleration of the moon around the Earth. If I know the distance of the moon from the Earth, and I know its acceleration, I can know its velocity, And when I know its velocity, I can calculate at that distance how long it will take to go around the Earth. And if you plug in the numbers, which I've done on my piece of paper, but I don't feel like doing now, you can convince yourself that you'll find that the period of an orbit of the Moon is one month. So, if the Moon is being pulled towards the Earth by the same force that's pulling towards the Apple of Earth, Newton could predict that the orbit of the moon will take one month.

And of course, we all know, the orbit of the moon takes roughly one month, 28 or 29 days. And therefore, it is the same force that's pulling the moon towards the Earth as the apple. That's what makes it universal. And so it's quite important, not just that he explained the motion of planets around the sun by this force of gravity that's proportional of 1 over R squared. But then he shows, and anyone can show, that given the measured acceleration, therefore of an apple towards the surface of the Earth,

and knowing the radius of the Earth and the distance of the Moon from the Earth, you could predict that the Moon will take one month to orbit the Earth, and of course that prediction is manifested in observation. Well, now we cannot just use it to discover that there's a universal law of nature, but we can now discover something hidden about the universe.

Because we can turn this on its head, that if the force of gravity is G massive of the object tons of the sun, the mass of the planet tons of mass of the Sun, the mass of the planet tons of mass of the Sun over r squared. And we also know that this is mv squared over R because this is, well, okay. Then we can basically show that the prediction of this is that the velocity squared of planets around the sun should be proportional to strength of gravity

the mass of the sun over R. That's a prediction, of course, of Newton that explains Newton's law, but now it tells us specifically, remember, Kepler told us the velocity squared of planets goes like one over R and with some constant proportionality. Newton tells us the constants of proportionality

are the mass of the sun times the strength of gravity. Now, it turns out at Newton's time, you couldn't measure the strength of gravity because you have to have two known masses to measure the strength of gravity. It took a hundred years later for Cavendish in 1798, I think, to be the first one to create a very careful measuring apparatus that could measure the strength of gravity between two heavy cannonballs of known mass and therefore measure now what we know is Newton's constant. But if we know Newton's constant and we know the velocity of planets around the sun and their distance, we can weigh the sun. We can determine the mass of the sun. And that is the way we weigh the sun now. Here is a way to measure the mass of the sun.

We can look at the velocity squared of planets, or in this case the velocity of planets around the sun, and the velocity should fall like one over the square root of GM over R. And so the shape of this curve, GM over R, goes up and down depending on what the mass of the sun is. and when you fit the planets, and you can see what a beautiful fit it is, we can weigh the sun to exquisite accuracy. We find out the mass of the sun is 2 times 10 to the 30th kilograms.

In fact, the fit is so good that we could weigh the mass of the sun using this kind of curve fitting to better than one part in a million, but we actually can't do that. The reason is we can't measure the strength of gravity Newton's constant to one part in a million. It's not known at that level of accuracy. So the accuracy of our knowledge in the sun is limited by our ability to measure the strength of gravity.

But we can certainly measure better than one part in 10,000 or so. And we do that just by fitting that curve around and we measure the mass of the sun. Once again, in physics, if it works, copy it. So if we can measure the mass of the sun this way by looking at the orbits of planets around the sun, let's try and weigh the galaxy. Well, as I say, if it works, copy it.

Well, here's it. This isn't our galaxy. It's the Andromeda galaxy, the nearest large galaxy are owned. But it looks like our galaxy. And in our galaxy, the sun is in a boring suburb on the outskirts, moving around the galaxy as the galaxy is orbiting. And the sun is moving around the galaxy at a speed of about 200 kilometers per second, orbiting once every 200 million years or so.

You can see our galaxy as seen from space this was taken with the Kobe satellite in 1989. But you can see our galaxy is a spiral galaxy and we're on the edge and things are moving around. And we can try and weigh the galaxy. Since we're almost at the outer edge of our galaxy, all of the mass of our galaxy is being used to pull us in that orbit. So if we know our speed and our distance, we can weigh the galaxy. And so let's do it. So let's do it. We have our speed about 200 kilometers per second, a distance, okay?

And we do that, we find out the mass of our galaxy is roughly 100 billion times the mass of the sun. And that's pretty good because when we look at it, there are roughly 100 billion stars. Everything works perfectly. But then we say, well look, maybe we can do better just like we did for the planets by looking at objects that are further away from the center of our galaxy. Since all the light is contained within us, all the mass should be contained. within it and therefore we can do exactly as we did for the sun we can do this Keplerian prediction that the velocity goes like one over the square root of

distance as you go further and further out and the first person whether there were a group of people who first did this but one of the people who did it most definitively looking at at many different objects like hydrogen gas clouds that were further out from the sun further out from the center of the galaxy in the sun and other other systems was Vera Rubin A remarkable astronomer, and I knew her. She was a friend of mine. She's passed away now.

Now there's a telescope named after her. And she was actually began her studies at a time when you couldn't be a woman and study in a place like Princeton in graduate school. She did night classes, I think, at George Washington University, I think. And eventually, anyway, was one of the people who first discovered something remarkable. If we look at other objects that are further and further away from the galaxy, molecular clouds, globular clusters, small groups of stars that are further and further away from the sun, satellite galaxies, all sorts of different things. What we find is we would expect their velocity around the center of the galaxy to fall like this,

because all of the mass within the galaxy, at least all the light, is contained within this distance. The sun is at the outer edge of our galaxy almost. But instead of falling off, it remained constant. This was a shock. Now, there's only two possible explanations of this. One is that gravity somehow behaves differently in our galaxy, which seems ridiculous.

Or the only way to explain why the velocity does not fall off is somehow is that the mass of our galaxy is increasing. More and more masses being contained within farther and further distances. And in fact, since you would expect the velocity squared to fall off as one over our, If the velocity remains constant, the only way you can understand that is if the mass is increasing linearly with distance. That somehow there's more and more mass out to distances of ten times as far from the center of the galaxy as our sun. This would suggest there's ten times as much mass. This was the first observation of what is now inferred and now known as dark matter.

And Verer Rubin and our colleagues were the first ones to try and definitively argue, that our galaxy was dominated by stuff that doesn't shine. Because remember all the light is contained within this distance. Out here, there's not much light at all. Therefore, and not many stars, therefore, and therefore we didn't think there would be much mass, but in fact, it was suggested that there's at least 10 times as much mass around the Milky Way galaxy as contained within the visible light region.

Now, this is a dramatic claim, and as Carl Sagan and others would want, say extraordinary claims require extraordinary evidence. And for a long time this wasn't believed. Because this is astronomy and astrophysics. It's hard to make measurements. There's lots of ways to get systematic uncertainties and systematic errors in there. And so these observations were first made in the 1970s,

and many people were quite suspicious of this. But over time, it turns out this is ubiquitous. Every galaxy we can see has what's called a flat rotation curve. Instead of falling off with the the light falls off, it's constant. And I'll just show you. Here is the function of distance from the center of the galaxy is the density of light in these galaxies.

If you assume the mass follows the light, this is the prediction you would expect from Newton for the velocity of rotation of objects around the galaxy, and you can see in both galaxies, it's constant. And every single galaxy, essentially spiral galaxy, we can see, we have the same phenomena. It appears that not just our galaxy, but all galaxies are dominated by stuff that doesn't shine. Now, maybe this isn't too surprising because there's lots of things it could be.

It could be snowballs. It could be cosmologists. Most people don't shine. We could fill it up with people and they wouldn't shine. Stars shine. But actually, as we'll talk about it perhaps in a later discussion, there's so much of this dark matter that we know it cannot be accounted for by stuff made of the same stuff as you. and me. There aren't enough protons and neutrons which make you and me up in the universe to account by a factor of at least five for the amount of dark matter orbiting galaxies,

and in fact also the same thing holds for clusters of galaxies, and there's just not enough normal matter. So we are convinced, well, there's two possibilities. Maybe Newton's law of gravity breaks down on the scale of galaxies or larger, and some people have been looking at this. It seems ridiculous from a fundamental point of view. Why should on the scale of galaxies suddenly Newton's famous law break down? That seems crazy. The other option is that there's something else out there that isn't made of the same stuff as you and I, and that's not so crazy, as I'll try and argue in later episodes.

It's quite reasonable to expect in the early universe new forms of elementary particles to exist that don't shine, and at least as abundant as normal matter. It still remains a mystery. for 40 years, I started 40 years ago proposing experiments to look for dark matter, and those experiments are ongoing underground now, and we still haven't seen it, but that doesn't mean it's not there. We still don't know the nature of dark matter, but if dark matter is made of a new type of elementary particle,

we'll either discover it in these underground experiments, or maybe we'll produce it in an elementary particle accelerator like CERN. It is one of the biggest mysteries, of course, in science, and it's discovered simply by applying the laws that Galileo and Newton derived several hundred years ago. But you see, just by a careful consideration of motion going from position to velocity, in now one hour,

we have now gone from what seems like a tedious and uninteresting discretion of how things move around the earth to discovering the hidden nature of the universe. And the unexpected. expected payoffs of a bold and simple change in view of motion have been profound. Understanding motion has led us to discover, at least, that most of the universe is hidden from our view. An amazing discovery and one that still is a mystery today. Just by a careful consideration, thinking outside the box, going from position to velocity.

Once we go from velocity, we can think of change of velocity to acceleration. Once you think of acceleration, you have Newton, and Newton discovering the nature of acceleration of orbiting objects discovers the law of gravity. And then applying that law of gravity to weigh objects tells us that the universe is far heavier than it should be. An amazing fact, and once again, it demonstrates what I always try and show

is that even elementary physics, which often seems boring when you first take it, because it doesn't seem to apply to anything interesting, can, within one lecture, apply to the nature of the universe and reveal some of its most profound mysteries. Okay, thanks. Hi, it's Lawrence again. As the Origins podcast continues to reach millions of people around the world,

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