11. Momentum & Impulse

Completely Inelastic Collisions

# 2D Collision

Patrick Ford

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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 an elastic collision. And we're gonna calculate the magnitude and direction of the final velocity after colliding. So let's get into the steps, we're gonna write some, we're gonna draw some diagrams from before and after. So here's what's going on here. So I've got this car like this and this car has a mass of 1000 and it's traveling east at 20 m per second. So in other words, it's a velocity kind of looks like this and I'm gonna call this basically this is car A so it's via is going to equal 20 m per second. Now, what's happening here is that it slams into a truck that's moving north. So what I'm gonna do is I'm gonna draw a little X and y axis like this. Alright, so I've got this velocity here, this truck has a mass of 3000 and it's moving totally north, right? So basically vertically up like that, that's north and that's east. So its velocity vector is going to look like this. I'm going to call this one be And this VB here is equal to 10 m/s. 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, trucks, whatever going to stick together, They're going to have a combined mass of 4000, right? 2000 3000. And then they're gonna move off in which direction? Well, you imagine this, right? If you have a car that's moving like this and a truck this movie 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 the diagonal. So here's what happens here. This truck is is going to go off. These two cars are gonna go off like this and this is my V final. And that's what I want to calculate when you calculate the magnitude of that vector. But I also want to calculate the direction as well. Right? So I want to also want to calculate the angle with respect to the X axis. Okay, that's really what we're trying to calculate here is V final and theta. So how do we do that when we actually just go back to using using, you know, normal vectors. If I want to calculate the magnitude of this vector, then I need its components. Right? I'm gonna need basically the X components and I also need the Y component. So the tricky thing here is that I want to calculate the final but I have to first calculate with Vfx and V F Y r. How do we do that? Let's move on to the second step here, which is going to write momentum conservation equations. So what I've got here is I've got M a v a initial plus, MB vb initial equal. And then remember these two things combined so we can combine their masses and sort of use that shortcut equation uh an A plus and B times the final. Now, are we done here? Well, actually, no, because for all the all the collisions that we've seen so far, we've only had them in one dimension and one access. We've had a car moving to the right and truck moving to the left or whatever to objects that 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 gonna write momentum conservation equations. But now we're just gonna do it for both the X and the y axis. It's just like any other two D problem that we've encountered when we did motion in two D. When we did forces in two D. We basically just wrote a bunch of equations in the X and the y axis, We broke them down and then solve them separately. That's exactly what we're gonna do here with momentum. So what I'm gonna do is I'm going to write equations for the X and Y. Conservation of momentum. So this is going to be a the initial plus and bVB initial equal M A plus MB V final. But the only thing that's different here is that I'm gonna conclude this is the X axis and this is the y axis and all the velocities are sort of implied that you're just gonna 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 anyway, remember that's just the mass of the car. That's going to be the 1000 now, V A X initial. So in other words, the velocity components of this car that lies along the X axis. Well this just lies like this. So it's basically just gonna be 20 like that. Now the second term, this is going to be M V which is 3000 but what about the velocity in the X axis? Remember this truck is still moving at 10 m per second, It's still going but it doesn't have a component in the X axis. Right? That the component of the velocity is only in the y axis here. So basically what happens here is that there is no X components and it's zero. So that just means that we can actually just cancel it out in our equation because it goes away on the right side. We're just gonna combine the masses. Remember this is 4000 V final X. So that's how you calculate the V final, basically what you're gonna get here, is that 20,000 is equal to 4000 the final X. And so your V final X is going to be five m per second. Alright, So now that we have that component here, basically what we can do is we can say this final, this v final is going to be the square root of V final X squared plus V final y squared. So, we just need those two components, like we said. Uh So when we just have this one now, we just need to go ahead and calculate the V final squared And to do that, we're gonna look at the Y-axis, we're gonna do the same music thing, right? So it's gonna be 1000. What about the via initial in the Y axis via remember just points along this, right? It's just 20 m per second to the right. What's the Y component of this velocity? It's zero. It's just kind of like what happened to the VB over here. So this is going to be zero here. So we can cancel this house And this is going to be plus Times 10. This is going to equal 4000 times the final in the Y axis. So what you're gonna get here is you're gonna get 30,000 equals 4000 V final Y. And you get that V final Y is equal to 7500 I'm sorry. 7.5, 7.5 m/s. All right. So now that we've got the two components of here, of these the v final. Now we can just go ahead and put them together and actually calculate the v final here. So this v final, I'm just going to bring this down here. The final is going to be the square roots of the five squared plus 7.5 squared. And if you work this out, what you're gonna get here is you're going to get 9. m/s. All right? So that's the one term. This is going to be 9.5 m/s. Sorry, 9.01. I'm sorry. 9.01. Now, what about the angle? Well, to get the angle we just use basically the inverse tangent, Right? So it's gonna be the tangent inverse Of the Y component over the x component. So in other words, we figured out that this was equal to five, this was equal to 7.5. And so this is going to be seven point 5/5. And when you work this out, what you're gonna say what you're gonna get is 56.3 and that's gonna be degrees 56.3°. And that is your final answer. Right? So these are both your numbers 56.3°. Alright, guys, so that's this one, let me know if you have any questions.

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