ï»¿ Alright, let's talk about something else. Here's the sun. Here you are standing on the earth. Some of that sunlight is going to come and hit you. Right, you're outside, you're absorbing sunlight. When you absorb sunlight what happens to you? You guys have all done this experiment, sit out in the sun, what happens? You get hot, right? You get warm. It heats you up. But let's ask a follow-up question. Does it also push you? We know it heats you up, but does it also push on you? Well. doesn't seem like it. I go out in the sunshine I don't get pushed over by the sunshine, right? But, in fact, it is pushing on you. Not very much but a little bit. And so there's this idea which we call radiation pressure. Radiation just refers to electromagnetic waves, and those waves, when they hit something, not only do they transfer energy to it and heat it up but they also push on it. So if I think about an absorber and here comes a wave and it hits the absorber and it gets absorbed in the material, if this is the before picture, then after, this thing starts moving. The wave is gone, it got absorbed. The object now is moving, it gets pushed away. So there is a momentum transfer between these two. And we need to understand exactly how that works. So, for an absorber, the momentum transfer from the wave to the absorber looks like the following. Delta p is delta u over c. How much energy was absorbed divided by how fast it was going. Speed of light. And this is the delta p that is picked up by that absorber. If it's reflected then that wave doesn't just get absorbed in the material, it actually bounces off and goes back the way it came. And if it's a reflector, now the delta p is twice what it is in the absorbing case. This is kind of like when we talked about throwing the super ball at the wall, it transfers twice its momentum to the wall. Whereas when you throw the sticky wad of clay at the wall it just transfers one of its mv to the wall. Okay, but if we have a delta p here, we have some change in the momentum, then we have to have a force because we know that force F is related to momentum. It's delta p over delta t. Alright, if I have a force and I have some area then I have to have a pressure. And pressure we write with this sort of capital P looking thing. Pressure is just force divided by area. Force, we just said, was delta p over delta t, but I know what delta p is. Delta p, in the absorber case, is just delta u over c. So this becomes one over A delta t, and then I have delta p which is delta u over c. And now let's rearrange this a little bit. And we can write it as the following: 1 over c times delta u over A delta t. But we know exactly what this stuff is. This is our good old intensity. And so this whole thing simplifies to that. Where S bar is our average intensity. Okay, so for an absorber, P is just S bar over C. For a reflector, and this is supposed to be capital P right, pressure, is 2 S bar over c. Let's take a look at an example. And this is something that people have actually undertaken. Here's the earth. There's the sun way over here. The sun is spitting out sunlight in all directions, some of that, some sunlight is skimming past our atmosphere. Let's take a spaceship, send it up into outer space, And inside that spaceship let's include a sheet of Teflon foil, or mylar that expands when it gets into outer space. This is something called a solar sail. Mylar is like a big reflector and so if that little spaceship has a big sail on it and we'll attach it to our little spaceship, put some antennas on it, the sunlight that's gonna come in hits the solar sail and pushes the spacecraft. And this is a real project that humans have undertaken, right? This idea of a solar sail. Let's see if we can calculate what the accelerating force is. Let's see if we can calculate that. Well, we know we've got pressure, right? We just talked about pressure right here for a reflector. Pressure is equal to 2 S bar over c. If I want to calculate the force, force is just pressure times area. And so I need to know the area of this solar sail. And they actually make these things big. Right, these solar sails can be 50 meters by 50 meters. And you want a big sail. Do we know what S bar is? Well, S bar is related to this electromagnetic wave that's coming in. And so S bar is the average intensity of sunlight basically where the earth is located. And we know what that is. It's roughly 1,400 watts per square meter. 1,400 watts per square meter. You know that it's a big number because when you park your car in the sunshine, you leave all the windows rolled up, it gets really hot in there. Because your windows have an area of, like, 1 square meter. And so they're letting in 1,400 watts of sunlight into your car. That's like you put the hairdryer on full-blast and left it in your car, it would heat up in there quite a bit. Okay, so let's see if we can calculate this now. We've got two times S bar which we just said is about 1,400 watts per square meter, we've got the area of the solar sail which is 50 meters squared, and then we're going to divide by the speed of light, 3 times 10 to the 8 meters per second. So somebody punch that into your calculator, tell me what you get. I'll just let you guys do that. While you do that I'll check Wolfram Alpha, if you're at home and you want to take a look, Wolfram Alpha is a great place to go. 2 times 1,400 times 50 squared all over 3 times 10 to the 8th. And I got 0.023. Is that what you guys got? So 0.023, and that 3 continues but that's fine. And the units are Newtons. Okay so that's the force on this sail. It doesn't seem like a lot of force, right? That seems like a pretty small number. Let's put this into comparison of something we know. What we know is that a quarter pound is about one Newton. So if I have 0.023 Newtons and I want to convert that to pounds, all I have to do is multiply by 1. So it's a quarter pound over one Newton, and so I get 0.023 over 4 pounds, and 0.023 over 4 is what? Well it's about 0.005 pounds. It's something really very small. Right, this is about the weight of a marble. A very small marble has this much weight. So, why would you do this? Right, if you're pushing on this giant spacecraft with the weight of a marble, who cares? Right, the acceleration of the thing is going to be really really slow. Why would you even do this experiment? Because sunlight is free, you don't have to send it any fuel with the rocket, it's continuous, it's on all the time, and in outer space there's essentially no resistance. There's nothing slowing you down. So even though this force is very small which means the acceleration is going to be pretty small, it keeps accelerating forever until it leaves.