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Two cars collide at an icy intersection and stick together afterward_ The first car has mass of 1050 kg and was approaching at 00 mfs due south The second car has m...

Question

Two cars collide at an icy intersection and stick together afterward_ The first car has mass of 1050 kg and was approaching at 00 mfs due south The second car has mass of 850 kg and was approaching at 25.0 m/s due west:Calculate the final velocity (magnitude in m/s and direction in degrees counterclockwise from the west) of the cars (Note that since both cars have an initiab velocity, You cannot use the equations for conservation of momentum along the X-axis and axis instead, you must look for o

Two cars collide at an icy intersection and stick together afterward_ The first car has mass of 1050 kg and was approaching at 00 mfs due south The second car has mass of 850 kg and was approaching at 25.0 m/s due west: Calculate the final velocity (magnitude in m/s and direction in degrees counterclockwise from the west) of the cars (Note that since both cars have an initiab velocity, You cannot use the equations for conservation of momentum along the X-axis and axis instead, you must look for other simplifying aspects:) magnitude m/s direction counterclockwise from west How much kinetic energy (in J) is lost in the collision? (This energy goes into deformation of the cars:) Additional Materials Reading



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Two cars collide at an icy intersection and stick together afterward. The first car has a mass of $1200 \mathrm{kg}$ and is approaching at $8.00 \mathrm{m} / \mathrm{s}$ due south. The second car has a mass of $850 \mathrm{kg}$ and is approaching at $17.0 \mathrm{m} / \mathrm{s}$ due west. (a) Calculate the final velocity (magnitude and direction) of the cars. (b) How much kinetic energy is lost in the collision? (This energy goes into deformation of the cars.) Note that because both cars have an initial velocity, you cannot use the equations for conservation of momentum along the $x$ -axis and $y$ -axis; instead, you must look for other simplifying aspects.

In this problem. You have one car traveling south with a 1200 kilogram mess and a speed of eight meters per second and another car traveling west at 17 meters per second with a massive 850 kilograms. And it tells us that these two cars collide and then travel off at an angle. So this is going to be a perfectly in the last two collision. But remember that moment, um, are vectors, so we can actually treat this as vectors before anything else so we can solve for our mo mentum in the UAE direction, we can assault for our momentum in the ex direction and then we can use that to determine our initial mo mentum. It's at an angle, and that momentum at the beginning at an angle would have to be equal to this moment at the end when they have collided so we can say our moment in the ex direction squared, plus our moment of the Y direction squared Square rooted will give us this moment. We're gonna kind of do this someone big Chuck. So our momentum of the ex direction is just 17 times 8 50 mass times velocity I swear. Plus that 1200 times eight squared gives us a momentum of 17,348 kilograms meters per second. Again, this is our moment before and after. And since it tells us they stick together than this, momentum is equal going to be equal to mass times velocity. So are mass is the total mess. So that's that 17,348 divided by my total mouse of 1200 plus 8 50 gives me velocity in this direction of about 8.46 meters per second. Now it also wants us this offer the direction of this velocity. So our direction of this velocity will be in the same direction as its moment. Um so we can actually go back and use moment for this. So the inverse tangent of my why component of momentum divided by my ex component of momentum. So again, inverse tensions opposite over adjacent gives me an angle of 33.6 degrees. Now this angle's measured from 180 so that's 33.6 degrees downward. If this is 180 degrees. So if we were to measure from zero that would be 1 80 plus 3 36 3 33.6 And the next part asks us to determine how much kinetic energy is lost. So for that, we simply need to calculate my kinetic energy at the beginning, my kinetic energy at the end, and then compare those two values. So my kinetic energy at the beginning, because kinetic energy is a scaler we don't need to include direction doesn't matter. That one is traveling South and one's traveling list. We can just calculate its total kinetic energy. And I get a kinetic energy of 161,000 225 jewels and then at the end there, both traveling together so I can add their masses together and then use the velocity that I found that 800 that 8.46 and this gives me a kinetic energy of 73,360. And then the difference between these two kinetic energies my final minus initial will give me in negative 87,000 800 and 64.1. Jules

Okay, so we know that a 1000 kg car who's traveling eastbound at 10 m per second, it hits a stalled 750 kg car, which after the collision, and we are told that this seven year 50 kg car goes 30 degrees north and east of four m per second. And we want to find the resulting magnitude and direction of the 1000 kg car after the collision. So we know that the initial momentum is going to be a mass times velocity of the movie car, which is just 2000 types, 10 to 10. That was in kilograms meters precipitate. So we know that the end, this is going east. The momentum has direction, which is straight east. So we know that the final momentum will also be 10,000 kg meters per second in the east direction. So, you want to break down the 750 kg resulting velocity into both directions. You know that it's gonna have a velocity in the Y direction and velocity the extraction. And we can saw those intrigue. The velocity in the Y direction will be mhm for sign of 30. And the velocity next directional beauty for of course under 30. And we get these numbers to be for six and Mhm. Two. Yeah. So now we want to calculate the momentum that the 750 kg car has. So in the ac direction, it has the momentum of 3.46 times 750 kilograms. Yeah. Which gives us an answer of 2005, 95 And in the Y direction it has a momentum of two times 7 50 which is equal to 1300 kilograms meters per second. So we know that initially we calculated that had a momentum of 10,000 kg per second going straight eastwards. Therefore, the resulting 1000 kg card, we'll have to have gone somewhere along the lines of lips because it has to cancel out the vertical component of momentum that the 750 kg card had as initially there was no vertical component of momentum. So we know that the car had a horizontal component of 2595. And we know that the momentum initial was 10 1000 has to equal the final, which we calculate here with 202,000 595 But the momentum of the 1000 kg car, so we know plus the plus the mass, which is 1000 times the velocity in the X direction of this card. So if we move, if we saw, if we isolate the velocity extraction, Yeah, we can do 10,000 minus 2595. We get 7405 is equal to 1000 times velocity and the extraction we saw for loss in the extraction and we get 10.405 So we know that it's going to be moving Yeah, man to the right, at a velocity of 7.45 the angle is unknown. Yeah. So, and this was in the extraction. So we know that momentum in the Y direction was initially zero, but this 750 kg car had a momentum of 1500. So if momentum in the Y direction initial was zero, then we know that we went to the final in the Y direction will also be zero. So 1500 will be equal to the velocity will be the equal to the momentum in the Y direction of the 1000 kg car. So we know this will be 1300 equals 1000 time velocity in the Y direction. Then we get that the velocity is 1.5 units per second. So we can have this and we know that it will have a resulting using a squared plus B squared it will have a speed of 7.55 m per second downward. We can solve the angle using inverse tangent opposite over adjacent 1.5. To write by 7.45 goes an angle of 11 point four or five degrees and this angle will be 11.45 degrees south of east. So that is part a part B. Now, I want to find the ratio of kinetic energy. We know that the initial kinetic energy will just be one half of the squared one half times 1000 kg times 10 squared which is 100 times I was invited to. And 50,000 angels. Yeah. You know the energy final will be the resulting speed we said was one half 750 kg times the speed given to us of four squared plus one half 3000 kg card. With a speed that we calculated of 7.55 square. We can calculate that out. And we find that it has a final energy of 63,000 2.5. So the ratio of energy final over Energy initial 63 I was in 2.5 over 50,000. Okay. Equal to 1.26 Okay.

Yeah. Hi when this question was given that a 7500 kg truck and It's a movement at five minutes of second Collides with the 15,000 kg car moving at 20 in a second And the direction of the car is 20 m of the second south of west. So that the south, this is west, south or west to be going south from west, which will be in this direction. And this is texas degrees. Just trying to talk. That's hard to create. And we want to find the final velocity and direction after collusion. And I'm sorry I said this was west when this actually east. So let me redraw this. So this is Yeah. Where? Yeah, this is south, so that's a degrees from west southwest 13. And with the result of course the factors for this. Mhm. And so this place is going to be, I told her the velocity here is This is for the car, for the 1000 parent regular Graham Car. And this is 20, it's just a second. So this side Going to be 20 sign that he understood is going to be trying to cost Archie. Yeah, I learned it. This car collides with a truck moving east. So we have a truck moving in this direction And he's moving at five m/s. And so the winter resolve due to the initial and final momentum for each and the vertical and horizontal directions. So first of all, let's take horizontal direction. I am the direction. Yeah. And from a lot of momentum. P musical to mass. Thanks for lost. So for the for the truck, the mass is 7500 times the velocity is fired and we're taking right and down. I suppose it'd directions. So this is right. Which is positive. And plus The mass of the car is 1500 times velocity or isn't our lifestyle the car which is Which is this year 20-20 or 30 because it is negative going to the negative direction. Okay. Mhm. Yeah. And the final, the last week, quarter final loss to watch so that they stick together so well then she do you have a common velocity V did you go through In Brackets? 7 5 +15 and we want to find people. So this is just this is just the X. Component to Vieques person. And calculating this, we get V. X. Is equal to and our velocity here comes down to one point to eat meters per second. So since it is positive, it means it is in the in the right direction. Mhm. Now for the for the for the vertical direction. Uh huh. The mass of the truck 7500 times the last in the direction of the truck. And that zero because it has only was moving east. So it has no critical components Plus one of fire times. 20 scientific. Fantastic. 20 sign 30. It's just just and difficult to B. Y. Times 7700 Plus 1500. We get a few wires calls. Yeah. 1.67 media access. Is this positive remain systems in the down direction. So to find the result on speed used by to grab the arm and good. I'm going to be skirt of E. X. Squared plus one squared visibly called get squared 1.2 Squared. Almost 1.67 sweat. Yeah. Should be equal to. And our result is going to be two points. One make that the second not for direction tender is you go to sign and bus of you. Y over X. And it'll be turned and birth. Well, yeah, 1.67 over one point to it. Great courtroom And the uncle gets is 52.5 degrees to remember were result of or philosophy into the town and actually right component. So this angle is going to be this this this angle will be 52.25 from the horizontal. So this is 2.5 the greatest and this would be 2.1 m/s. This is the final velocity of the system.

Yeah. Once again, welcome to a new problem. This time we have a truck and a sports car. So the sports car were given that the sports car is moving in the, uh towards the left eastward, so that means it has a negative velocity. The mass of the sports car happens to be given us 1000 and 50 kg, and it's moving with an initial velocity, um, towards the left, which is negative 15 m per second. That's the sports car. And then we also have a truck. Well, given that you know there's a truck and it's moving towards the right, the mass of the truck happens to be, um, 6000. So the mass of the truck happens to be 6000 320 kg. The velocity initial velocity of the truck is given us mhm 10 meters per second. So that's what's going to happen. That's the information we're given its positive because it's moving towards the right. Um, we want to find in part a the a final velocity when the truck and the sport's got collide with each other. So you know they're gonna collide and and stick together. Um, and we don't know what direction they'll go, but, you know, you can assume they're going towards one direction. Uh, and we want to find that final velocity when the collision happens. Uh, the second thing we want to find out. So this final velocity we want to find the magnitude and direction. Those two things you want to find out is going to go to the left or will it go to the right? The second thing we want to find out in part B is the speed that the truck is moving. Um, so the initial speed of the truck such that Okay, such that, uh, they stop on collision. So another way of framing the problem would be if they both have an initial, uh, the same the same momentum. Um, when they're moving towards each other, they're gonna stop, and that's what be. And then in Patsy, we want to find what we call the or the change in kinetic energy for the two parts. So, you know, what's the what's the change in kinetic energy in part? A. And then what's the change in kinetic energy? Uh, in part B. So that's what we want to find out. And then the final thing, Uh, the asking is which one is bigger? So is the change in kinetic energy in part A Mm. You know, Is the change input a greater than in part B, for example? You know, that's that's one thing you want to, uh you want to check? Okay. Yeah. Mhm. Yeah. Mhm. Yeah, sort of. Yeah. So, um, that's the information we're given in the problem. And, uh, you know the first part The first part of the problem is that we are going to compute the final velocity using the law of conservation of momentum, using the law of conservation of momentum. We get to see that the initial momentum, it goes to the final momentum. We can switch that because we are targeting the final velocity. So they're sticking together. So the moss off the truck, plus the mass of the car and the A final velocity is the same as, um is the same as the mass of the truck and the initial velocity of the truck. Uh huh. Plus the mask of the car, times the initial velocity of the car. And so, you know, moving things around to see that the final velocity is the initial momentum. Mhm of the car, plus the final momentum off the car over the two. Okay, the sum of the two masses together. And so you know, we'll we'll do this in the next page. So the final velocity becomes 6320 kg times. That's the momentum of the truck. 10 m per second, plus the momentum of the car. The sports car, um, times negative. 15 m per second. You want to divide that by the some of the masters, which is six, 6320 kg plus 10 50 kg. Um, and so you know your final answer become six point 43 me this second, and it's positive. So the if you go back, we're looking for the magnitude and direction. So it means that the, um two, uh two. Yeah, vehicles will move towards the right wind. A common velocity. Oh, 6.43 So we'll see 6.43 m per second. You know, that's what's going to happen to that. Uh, the second part you can see, we were We wanted to find the speed such that the, um So they want to find the speed of the truck if they happen to collide and stop. So at what speech? Had the truck been moving so that they both, uh, they stopped after collision. So we'll do that on the next page. So in terms of what we're saying is we have a truck and then we have a car. Um and, you know, they were moving towards each other, they collide, and then they stopped. So if they collide and stop, that just means that the momentum, which is the mass of the truck and the initial velocity of the truck, has to be equal to the mass of the car. Um, and the initial velocity of the car. So we don't know what this velocity is. Uh, but we we can solve it. We put those two together. Remember, this is the outcome. When they stick to each other, it means the momentum's are the same. Um and so, uh, this turns out to be the mass of the car is 10 50 kg and its initial of lost eight towards the left is negative 15 meters for second. And and the mass of the truck is huge. It's 6000 320 kg. That just means that the new velocity happens to be negative 2.49 m per second. Uh, it's you gonna be moving towards the left. So if you go back, we were saying, you know, what's the speed such? That they're going to stop? You know, that's what you're looking for. Speed has no magnitude or speed has a magnitude no direction. So it's just gonna be 2.49 m. Second, that's the speed of the truck if they have to stop if they have to stop when they collide. So it means that you know, the the, uh, in terms of velocity the truck will be moving, You know, that week before they stop. Um, and that just comes from the arrangement organization of the problem. The last part of the problem, which is Patsy. We're going to do that in the next page and was saying that you know, what's the change in kinetic energy in the two parts? What's the change in kinetic energy? Remember, this is what happened in part a. So change in kinetic energy is, uh, the final kinetic energy of the system, minus the initial kinetic energy of the system. The final kinetic energy of the system. Remember that when they collide, they stick to each other. So that's M C. And the final squared minus the initial, which is before they stick to each other. This is the initial momentum of the truck. Um, and then plus, you know, we need a parenthesis there just to make our life easier, plus the initial momentum of the car like that. And so, uh, when we plug in the numbers, we get one half, 6000 320 kg, plus the mass of the car is 10 50 kg times the final velocity that we computed was 6.43 m per second. Um, and if you can recall, it was actually positive. So 6.43 m per second square. That and then we subtract. One half mass of truck is 6000 320 kg, and the velocity of the truck initial velocity of the truck, you can see was, um uh, the initial velocity was 10 m per second, and you still have to square that because we're dealing with, um, kinetic energy in this case. So 10 m per second squared, and then also we have plus one half mass of the truck is 10. 50 kilogram. The last year of the car is if we go back. He was negative. 15. So we have to include that. Meet us second squared. Um, and so the change in part A would be a positive to 81.7 times 10 to the three. Jules, you want to call that? Mm. You know, that's the change in kinetic energy. They're asking us to compute that find the change in kinetic energy of the system of the two in in both the situation. So, uh, this is the change in kinetic energy For the first part, which is a two 81 0.7 times 10 to the three. Jules, the second part, we have to get the change in kinetic energy once again. You know, it's the final, uh, minus the initial kinetic energy. They stick together. So the final kinetic energy zero jewels. Um, and we want to find the initial kinetic Kennedy before they stick together. So one half, uh, m t v t initial squared, plus one half m C VC initial squared. Those are the initial and final momentum in that case. So we have one half times the mass of the truck. 6320 kg. Always a good habit to have the units there. Initial of the law. Steel of the truck. Remember, this is the velocity that's going to make them stop when they when they collide with each other. And then this is plus one half the mass of the truck is 10 50 kg. The initial velocity in that case was negative. 15 m per second squared. When you simplify that, you get negative 1 37.7 times 10 to the three jewels. You know, that's the change in kinetic energy in the second case. Okay, Mhm. And so we can conclude that, um, you know, as they were asking as part of the problem, you know, if you go back, you see that the change in kinetic energy in the first case, um, is greater than in the second case. So, in terms of magnitude, so we'll say the magnitude of kinetic energy in part a mhm, uh, is greater than in part B. Okay, Hope you enjoy the problem? A couple of steps going on the first part. We're finding the final velocity when they stick together. 6.43 m per second. The second part. We're finding the speed of the truck. The ideal speed of the truck for them to stop 2.49 m per second for the truck. This is the last day of the initial velocity of the trunk, as we call it. Well, actually, this is the initial blast in the truck. And then, you know, you can change that to speed. Speed is a scalar quantity. Um, and then finally you compare The energy is in part a n b. And you can see that the energy part is bigger. Hope you enjoy the problem. Feel free to ask any questions, uh, and have a wonderful day. Okay, Thanks. Bye.


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