Two vehicles are approaching an intersection. One is a 2500-kg pickup traveling at 14.0 m/s from east to west (the -x-direction), and the other is a 1500-kg sedan going from south to north (the +y-direction) at 23.0 m/s. (b) What are the magnitude and direction of the net momentum?

Video transcript

Hey everyone, Welcome back in this video. We have two trucks that are going to be driving. Were given some information about them and we want to determine the magnitude and direction of the total momentum. Alright, so, let's first start by drawing out the system. Now we're told that north and east is going to be the positive Y and X directions. Okay, So we'll take north to be up east to be to the right. And those are going to be our positive directions. Alright, now our system has two trucks. So we have the first truck here in red, We're told as a mass of 2200 kg We're also told its speed is 8.6 m from north to south. Okay, while we've said north is up, so going from north to south, this car is gonna be traveling down All right now because the car is traveling down and up is chosen to be a positive direction. Its velocity is going to be negative, so negative 8.6 m/s. All right now, we have the second car, we'll draw it in blue And we're told that it is eight, kg and it is headed east to west. Okay, so again, east is right. So going east to west, the car is going to be going to the left because it's going to the left and the negative extraction, the velocity is going to be negative. We're told the speed is 19 m per second, so that velocity is going to be nine x 19 m per second. Alright, great. So now we've drawn our system, Let's go ahead and start with this solution. Now, we're asked to find the total momentum. We need to recall that the total momentum is going to be made up of momentum in the extraction and the momentum in the Y direction. So let's go ahead and find the momentum in each of those directions. So we have the momentum in the X direction. Well, what's in our system? Remember that both of these cars are part of our system? Okay, So everything circled here is our system. So the momentum in the extraction is going to be the momentum of the first truck in the extraction plus the momentum of the second truck in the extraction. Now recall momentum is mass times velocity. So, for each of these terms we're gonna put the corresponding mass times the corresponding velocity. All right. Now, if we're looking at car one, Car one is moving up and down, I guess it's moving down. Okay, So it's not moving left to right, so it's X velocity is going to be zero. Okay, so that first term is just going to go to zero and then filling in the details for card to we're going to get 1980 kg times minus 19 m per second. Okay, Which is gonna give us a momentum in the extraction of minus 37, kg meters per second. Alright, so that is our momentum in the X direction. Okay, and now doing the same thing for the Y direction, the momentum in the Y direction again is going to be made up of the momentum in the Y direction of both truck one And Truck two. Okay. Mass times velocity for each of these. Now, this time when we look at car two, Car two does not have a wide velocity. Okay, Car two is moving to the left, it's not moving up and down. The Y velocity is going to be zero. So that second term goes to zero and we can work out the details of that first term, 2200 kg times minus 8.6 m per second. Okay, that gives us minus 18, kg meters per second. Okay, and that's our momentum in the Y direction. Alright, so now we have our momentum in the X direction and momentum in the Y direction. We're looking for our total momentum, the magnitude and direction. Okay, so let's draw a little sketch here, we're just gonna draw the access like this. Okay. And we know that our car is going or the total momentum story is going to the left because the X momentum is negative. Okay, right is positive, the X momentum is negative, so our total momentum going to the left and it's also going down. Okay, again, the momentum in the Y direction is negative. We've chosen up to be positive, so it's going to be going down as well. Okay, well the total momentum is this gonna be this line here. Okay, right here. So this value on the X direction, 37,620. Our ax momentum In the Y Direction 18,920. Well, how do we find this yellow line by the green theorem? We know how to do that. We'll go ahead and use by the green theorem so we can write that P squared. The total momentum magnitude of the total momentum is going to be P X squared plus B y squared. Ok. That's just pythagorean theorem, C squared equals a squared plus B squared. And if we fill in the values, we have 37,620 in the X direction 18,920 in the Y Direction. Okay. And if we work that out, oops, sorry, not P squared P is going to equal plus or minus 42, 0.75 kg meters per second. Okay. And in the question they're asking just for the magnitude, first magnitude and direction. So when we're talking magnitude we can go ahead and take the positive group. So 42,109 0.75. And if we want to write this in scientific notation we can move this decimal place 1234 spots. So that's going to be 4.21 times 10 to the four kg meters per second. And that is going to be our magnitude magnitude of the momentum. The last thing we need to do is to figure out the angle. Now we draw a diagram here. Remember that angle is measured from the positive X. Direction. So the angle we're looking for is this total angle here? Okay. Alright, so the angle theta Actually going to be equal to 180°. Okay, that's gonna be from the top two quadrants. So that whole top is 180° plus this little angle here, we'll call it theta prime. Okay. And data prime we can find because we have the X and Y directions of our momentum. Okay, so we know that we can say tan of data prime. Okay, so here tan opposite over the adjacent, so that's going to be 1809. 20 over 37,620. Okay, so fed up prime is gonna be the inverse tangent of that value And we get 26. 892°. Okay, now it's easy to think that this is our answer but remember we measured the angle from the positive extraction. So we have to go back to that total feta plug in the theta prime. We were getting And we get an answer of 206.7° for our angle theta. So that's going to correspond with solution number or letter B. Alright, great. Thanks for watching guys.