Momentum. Conservation of momentum. And a block sliding down an inclined plane. So we're getting back to some review of previous concepts as well as taking a look at how they combine with momentum. We did this problem before but I want to walk through it here just to make sure that we've all got it down. So the details, a 4.4 kilogram block, starts at the top of a 3 meter long incline that makes a 22 degree angle to the horizontal. All right. There's a second block at rest at the bottom of the incline that the 4.4 kilogram block collides with and sticks to, traveling together after this collision with a shared speed of 1.25 meters per second. Now the question here is as this block slides down the incline, it's got a coefficient of friction of 0.155 mu is going to be equal to that. And we're wondering what is the mass of this block here, the second block? All right. So I think we've got this one nice and translated. We can get rid of the problem here. And keep going from there. Let's take a look at the end here. We've got two blocks collied with each other, stick together, move is 1. That's a certain kind of collision we had a name for it before. And that was a perfectly inelastic collision, which means these two stick together and they slide together across the ground. And we know how fast they're moving at the end. All we need to do is figure out what the momentum is at the beginning and we can solve this part of the problem at the end. So what we're going to look for at the very end of this problem is using our conservation and momentum equation. And so when we look at this, we know M1, we are looking for M2. We don't know, actually we do know V2 because it says it started at rest up the top or at this point down at the bottom. So V2 is actually zero. That term goes away. We know then M1, we know VF. That's 1.25 meters per second. We don't know M2 and we're looking for that. So our unknown is right here and everything else in this equation we know except for that guy. So we need to figure out what is the velocity of this block right here, right before it collides into the other block. If we can find that, then we're golden. All we have to do is plug it into this equation and we're done. The question then becomes how do we get that going? How do we figure out this velocity? So we're going to look, think back to some of our previous units. We got a block sliding down a plane with a coefficient of friction. When it gets down to the bottom, it's going to be moving with some velocity. And so there's two ways to do this. I'm going to show you the first one first and we'll see if we'll go through the second one or not. But I'll explain at least with both of them are. And the first one is, well, either way, we're going to have to draw a force diagram and consider the forces. So let's get started doing that before we move into kinematics. All right. So here's our force diagram. We've got gravity pointing straight down, normal force pointing normal to the surface. So perpendicular to the surface. We've got the friction force which is pointing parallel to the surface. In this case, it's pointing up the incline because it's moving down the incline. Friction always points the opposite direction of motion. And so let's take a look here. The next thing we have to do is actually break up this component into that vector into components. All right. Now we've got our components. So we're going to go through and actually figure out what the values are for each of these. If we remember, fg is going to be equal to mg, the mass of the object times the force gravity. So that's 4.4 times 9.8, which comes out to be about 43 newtons. And then what we need to do is break this up into components further by using some sokatoa. If you remember, we went through this exhaustively before, but fg parallel is going to be equal to this number, force of gravity, times the sine of theta. So let's write these down. All right. So there we go. We've got these written down our components of these vectors. These two right here. The next step that we need to do is actually write down equations for the sum of the forces. In the perpendicular direction, we've got that the forces is the normal force pointing upwards in the perpendicular direction, minus the gravity force pointing downwards in the perpendicular direction. And since the block is not accelerating this direction, it's only accelerating down the hill, this is going to be equal to zero. So what that tells us is that the gravity force has to be equal to the normal force. So this here is equal to the perpendicular gravity force, but it's also equal to the normal force. With that settled, we can figure out what the friction force is in this case, because it's equal to mu times the normal force. Now that we've got our friction force, we can go ahead and do one more sum of forces, but in the parallel direction to figure out the acceleration of this object. And this is the expression we've got. That the parallel forces when all added together is the downhill minus the uphill force. That's equal to m times a. So if we're playing in our numbers for each of these and divide by the mass, we should find the acceleration. And we find that our acceleration is equal to 2.263 meters per second squared. Now we've got this. All that we have to do is use kinematics to figure out the velocity of it at the bottom of the ramp. So let's think about this. We know the acceleration. We know the initial velocity is zero. We know the distance that goes down the ramp and we're looking for the final velocity. The only one we're not caring about is time. There's only one of our kinematics equations that doesn't use the time. Let's see if you can remember which one it is as I'm erasing this. So the other one without the time, that's our v squared equation. So let's put that up here. And the initial velocity is zero, final velocity is what we're looking for. We just calculated the acceleration and the distance was given. So let's plug it in. So if we plug everything in, we get 13.576 meters squared per second squared is equal to the final velocity squared. So we take the squared to both sides. We get this 3.684 meters per second. That's the speed it is when it hits the other block. So we're going back to our equation we wrote down at the beginning to do the last step of finding the mass by plugging into this equation. It's going through doing what we had done before. We're going to do everything in here. We get that the mass 2 is equal to 8.56 kilograms. You've got to plug everything in, solve for m2. We do that by subtracting this to the other side, dividing both sides by 1.25. And that's it. That's how we can figure this out. If instead of using kinematics and finding the acceleration, you wanted to use energy and find the work, you could do it that way too. The things you would have to talk about is at the beginning we've got potential energy. We have our gravitational potential energy due to the height. So you'd have to calculate the height using sopetola. At the bottom we lose all that energy, but and it gets turned into kinetic energy. But some of it is lost to friction. So we find out the work done by friction, which would just be the force of friction, times the distance down the ramp. So 16.196 newtons times 3 meters would be the work done by friction, removing energy from the system. So we take the potential energy to beginning minus that number there, and that would be equal to the amount of kinetic energy down at the bottom. We set that equal to 1.5 mv squared, and we can calculate for the velocity. If you do it this way and do everything right, you should get the exact same final velocity. There are two different ways to do this same problem. But that's how we do it. It's two steps. At the beginning you've got an object going down the ramp at the end it collides and keeps on moving. So we're going to do this where we keep adding some extra things to the front and the back. It's a good review for the final as well as making these questions just a little more interesting. Hopefully we don't get tired of these blocks because we're going to be seeing a lot of blocks for some of these problems. So just a little bit left for this unit, and then that's it. Thanks.