Hey, everybody. So in this problem, we have a car and a truck that's colliding. Now luckily, nobody's hurt, but we do have the cars that stick together and lock together during the collision. So this is a completely inelastic collision, and we're going to calculate the magnitude and the direction of the final velocity after colliding. So let's get into the steps. We're going to write some, we're going to draw some diagrams from before and after. So here's what's going on here. I've got this car like this and this car has a mass of 1,000 and it's traveling east at 20 meters per second. So in other words, its velocity kind of looks like this and I'm going to call this basically this is car A. So its v_a is going to equal 20 meters per second. Now what's happening here is that it slams into a truck that's moving north. So what I'm going to do is I'm going to draw a little x and y axis like this. Alright? So I've got this velocity here. This truck has a mass of 3,000, and it's moving totally, north. Right? So, basically, vertically up like this. That's north and that's east. So its velocity vector is going to look like this. I'm going to call this v_b and this v_b here is equal to 10 meters per second. So once they lock together, what's going to happen? Well, if you draw the after diagram, you can kind of sort of intuitively guess what's going to happen. Afterwards, the two cars, the trucks, whatever, are going to stick together. They're going to have a combined mass of 4,000. Right? 1,000 plus 3,000. And then they're going to move off in which direction? Well, if you imagine this, right, if you have a car that's moving like this and a truck that's moving like this, once they stick together, they don't move totally to the right or perfectly up. They move sort of a combination of the two directions. They sort of go off in a diagonal. So here's what happens: This truck is going to go off. These two cars are going to go off like this, and this is my v final. And that's what I want to calculate. I want to calculate the magnitude of that vector, but I also want to calculate the direction as well. Right? So I also want to calculate the angle with respect to the x-axis. Okay? That's really what we're trying to calculate here: v final and theta. So how do we do that? Well, we actually just go back to, you know, using, you know, normal vectors. If I want to calculate the magnitude of this vector, then I need its components. Right? I'm going to need, basically, the x components and I also need the y components. So the tricky thing here is that I want to calculate v final, but I have to first calculate what v_fx and v_fy are. How do we do that? Well, let's move on to the second step here, which is we're going to write momentum conservation equations. So what I've got here is I've got m_av_a initial plus m_bv_b initial equals, and then remember, these two things combine, so we can combine their masses and sort of use that shortcut equation, m_a plus m_b times v final. Now are we done here? Well, actually, no. Because for all the collisions that we've seen so far, we've only had them in one dimension, on one axis. We've had a car moving to the right and truck moving to the left or whatever. Two objects are basically just moving along the line. But in this problem, we have a two-dimensional collision because one is moving sort of left and right and the other object is moving up and down. And here's the idea. For a two-dimensional collision, we're still going to write momentum conservation equations, but now we're just going to do it for both the x and the y-axis. It's just like any other 2d problem that we've encountered. When we did motion in 2d, when we did forces in 2d, We basically just wrote a bunch of equations in the x and the y-axis. We broke them down and then solved them separately. That's exactly what we're going to do here with momentum. So what I'm going to do is I'm going to write equations for the x and y conservation of momentum. So this is going to be m_av_ainitial plus m_bv_binitial equals m_a plus m_bv final. But what the only thing that's different here is that I'm going to include this is the x-axis and this is the y-axis, and all the velocities are sort of implied that you're just going to look at the x and y axes respectively. Alright? So then how do we do this? Well, let's just jump right into the x-axis because we're going to go ahead and plug values and solve. So n_a, remember, that's just the mass of the car. That's going to be 1,000. Now v_ax initial, so in other words, this the velocity component of this car that lies along the x-axis. Well, this just lies like this, so it's basically just going to be 20 like that. Now the second term, this is going to be m_b which is 3,000, but what about the velocity in the x-axis? Remember, this truck is still moving at 10 meters per second.