Hey, guys. So now that we've talked about velocities and gravitation, there's a very specific velocity that you're going to need to know called the escape velocity. Let's check it out. So basically, what the escape velocity is, it is the minimum launch speed that you need in order for an object to escape. And now what that word escape means, what does that mean? It means that this thing stops when it's very, very far away, and it can never come back towards the earth. So let's think about something for a second. So we're used to on earth, we'll throw an object upwards. It'll sort of reach the peak, and it'll come all the way back down because of the force of gravity. Right? We know that gravity pulls everything downwards. If you throw this thing a little bit harder, so we've got some initial right here if you throw this thing a little bit harder, it's still going to go up. Gravity is going to get a little bit weaker, but it's still eventually going to come crashing back down towards the Earth. The idea is that there is a launch speed. So there is a launch speed, a minimum launch speed that is so fast, so we're going to call that v_escape. Now what happens is it gets all the way up to this magically very, very far-away place. And what happens is that the force of gravity can't pull it back down towards the earth again. And so what happens is that it stops and never returns. Now we know, but when two objects get really far away from each other, the force of gravity approaches 0. So in other words, as this little r goes to infinity or gets really really really big, we know that the force of gravity is equal to g M m over r². So if this thing in the denominator gets really really big, then the force goes to 0. Okay? So what that means is that the object is going to stop, and when it stops very far away, the final velocity is equal to 0. So, in other words, v_final, when it gets all the way out here to this magical place that's infinitely far away, the final velocity is equal to 0. And the reason for that is that if it wasn't 0, if it were something that is greater than 0, then it wasn't the minimum launch speed that you could have thrown this with. So I'm going to make up a number here just for a second. Let's say you throw this up at, like, a 1000 meters per second, It gets up all the way out here, and the final velocity is equal to 5 meters per second. Well, then that means that you could have thrown it a little bit slower, and it still would have gotten out here with 0 meters per second. So that's what that means. You just have to throw it with the minimum launch speed. Alright? So this escape velocity here. And we know that we can't use kinematics because the accelerations are here, and we know that we can't use kinematics because the accelerations are constantly changing. So we have to use conservation of energy. So we've got initial kinetic and potential, plus any work done is equal to final kinetic and potential. Now, when you are throwing this object up with some initial velocity right here, we know that the kinetic energy is going to be 1/2 m v². So we've got math xmlns="http://www.w3.org/1998/Math/MathML"> 1 / 2 m v 2 , and now the initial gravitational potential energy is not 0, because you still have some distance away from the center of mass. So it's going to be negative g, big M, little m over little r. Now, we're talking about work. Sorry. We're talking about gravity. We know gravity is a conservative force. So that means there's no work done by nonconservative forces. It's kind of a double negative there. Now what about these last two? What about the energies when it finally reaches up this very, very far-away place? Well, we said that the energy or sorry. The velocity is equal to 0 when it finally gets out all the way over here. And we know that the kinetic energy is also 0 because it depends on this final velocity. Remember math xmlns="http://www.w3.org/1998/Math/MathML"> 1 / 2 m v 2 . So this thing is math xmlns="http://www.w3.org/1998/Math/MathML"> 1 / 2 m v final 2 , but we know that this thing is going to be equal to 0. So that means that the kinetic energy is going to be 0 there. Now what about the gravitational potential? Again, remember the equation for gravitational potential, it depends on that little r distance. So what happens is that as this distance gets really really big, then the gravitational potential energy will also go to 0. So that means that both of these things on the right will go to 0, and that allows us to figure out what the escape velocity is. So we're going to go a