A 2100 kg truck is traveling east through an intersection at 2.0 m/s when it is hit simultaneously from the side and the rear. (Some people have all the luck!) One car is a 1200 kg compact traveling north at 5.0 m/s . The other is a 1500 kg midsize traveling east at 10 m/s . The three vehicles become entangled and slide as one body. What are their speed and direction just after the collision?

Video transcript

Hey, everyone in this problem. A 400 g clay ball is initially moving to the West at 3.2 m per second. A 2nd 600 g clay ball traveling south at six m per second collides with the first clay ball and sticks to it. At the same time, an 800 g clay ball traveling west at eight m per second collides with the first two balls and sticks to them. All three clay balls stick together and move as a single unit. We're asked to determine the final speed and direction after the collision of these balls. We're given four answer choices. Option a 3.21 m per 2nd, 64.3 degrees west of South. Option B 6.64 m per second, 18.4 degrees south of West. Option C 8.32 m per 2nd, 45 degrees west of South. And option D 4.71 m per 2nd, 25.1 degrees south of West. OK. So let's draw a little diagram to get a sense of what's going on. And we have three time points that we really need to consider in our diagram. So we have the initial time point and we're just gonna call that TNO and at T knot, we have our 400 g clay ball moving West. So it's gonna be over here. We're gonna call this ball one. So it's gonna have mass one which is 400 g and it's traveling to the West. OK. And we're gonna take to the north and to the east as our positive direction. So up into the right. And so the velocity initially of this first ball is going to be equal to negative 3.2 m per second because it's moving in the negative direction. Then we have our second ball 600 g traveling south at six m per second. So we have this bow, it's traveling south. So it's traveling downwards, it has a mass and we're gonna call this M two for the second ball of 600 g, it's going downwards which is in our negative direction. So it's also gonna have a negative velocity and it has a velocity of six m per second. So it's gonna be negative six m per second. And then we have a third ball, OK? It's kind of on the outskirts. So it's over here somewhere and it is moving west at eight m per second. So it has a velocity of negative eight m per second, OK? In a mass of 800 g. So this is our T knot time point and we have these three balls, all of them separate, all of them moving in their own way. Then we have a collision and the first collision is between the 1st and 2nd clay balls. OK. So the first two clay balls are going to collide and they're going to stick together. We're gonna call this M 12, which is gonna be the mass of one plus the mass of two. OK? They're sticking together. So now we have this one total mass that's gonna be equal to g. We don't know which direction or they're moving or the speed that they're moving with. But we know they stick together and we know that that third ball isn't involved in the collision and it is still moving to the West with that initial speed of negative eight m per second or velocity. And it has the same mass it did before 800 g. OK? So this is our kind of intermediate time 0.2 of those balls collide stick together and we have the third one still moving. And then we have what we're gonna call our final time point. And that is when all three of these stick together. OK? So that last bowl collides with the other two that are already stuck together. And we get this total mass, we'll call it M T, which is the mass of all three of them combined because they're stuck together. And this total mass is going to be 1800 g. OK. 400 plus 600 plus 800 which gives us g. All right. So what we wanna figure out is the final speed and direction after this third or second collision? Ok. So at this third time point, what is the velocity, the speed and the direction? Now we have no net external forces acting on this system. So we have a conservation of momentum. Now, we can do a conservation of momentum between this initial time point and the second time point. Hm. That's gonna allow us to figure out how the first two balls stuck together are moving their speed, the direction. And then we can do a conservation of momentum between that and the third time point. But we actually don't need to use the intermediate step. We have enough information and because momentum is conserved between the first step and then the second step. And then the third step, we can just compare the first step and the last step. Hm. So what we're gonna do is we're gonna say that the momentum initially is equal to the momentum finally. OK. So this knot represents this T knot time point and the F represents this T F time point. Now we have motion in both the X and Y direction. So we need to consider the momentum in the X direction in the Y direction separately. So we're gonna start with the X direction. Now recall that momentum is going to be mass multiplied by velocity two. Now, when we're talking about momentum initially, what do we have to consider? Well, we have the momentum initially from ball one in the extraction, the momentum from ball two in the extraction and the momentum from ball three in the extraction. And the same in the final case, the momentum of all one, the momentum of ball two and the momentum of ball three. Now again, mass times velocity that's gonna give us our momentum. So for each of these terms, we get the corresponding mass multiplied by the velocity. So M one B 01 X plus M two V +02 XX plus M three V +03 X is equal to M one V F one X plus M two V F two X plus M three V F three X. OK? No, let's substitute in the values we know and see where that gets us. So F one is 400 g V 01 negative 3.2 m per second. OK. When we're talking about momentum, momentum is a vector. OK. So the velocity in this equation, we're talking about velocity, not speed. Velocity includes information about the direction. So we have to include the negative there. OK. All right. And it's that entire velocity because that first ball is moving in the west direction completely in the horizontal direction. So when we're talking about X, its entire motion is in that X direction now ball two is moving to the south. So its entire velocity is in the Y direction. So its X component of its velocity is actually zero. So that second term goes to zero and the third ball, the third ball is moving West. OK. So again, its velocity is going to be entirely in the X direction. So we have its mass 800 g plus it's a velocity negative eight m per second. And this is equal to the right hand side. Now, we have these three velocities that are in our equation at the end. But remember that these three balls all stick together. So they're gonna be moving with one velocity, they're all gonna be moving with the same velocity. And so we have the sum of the masses, we can factor out this velocity term because it's the same, we get 400 g plus 600 g plus 800 g multiplied by the final velocity in the X direction. And this is what we want to find. OK. We're looking for the final velocity. So we need the X component and the Y component. So let's solve for V FX. Now, if we simplify in the left hand side, we get negative grams meter per second. And on the right hand side, we have 1800 g multiplied by V F solving for V FX, we have gram times meter per second divided by gram. So we're left with a unit of meter per second, which is what we want for velocity. Let me get that this is equal to negative 4. repeated meters per second. Now, one thing I want to note here is we did convert grams to our standard unit of kilograms. OK? And the reason for that is when we were looking for the velocity, we ended up dividing a mass by another mass. So that unit of gram actually canceled it. OK. So that's why we didn't need to convert to that standard unit of kilograms. OK. But be careful because in some cases, if you're not doing that division, you will need to do that conversion. All right. So we have the X component of our velocity. Now let's switch over find the Y component and then we can find that total magnitude and direction of our motion. So the same thing in the y direction, the initial momentum in the wider direction is equal to the final momentum in the wider direction, the momentum is made up of the momentum of the three balls. So momentum initially of ball one in the wide direction plus momentum, initially, of ball two in the wide direction plus momentum of initially of ball three in the wide direction. Same thing on the right hand side, we have P F one Y plus P F two Y plus P F three Y. And again, the momentum is equal to the mass multiplied by the velocity. So we can sub that in for each of these terms M one V, not one, Y plus M two V +02, Y plus M three V +03 Y that is equal to M one. And before I go further here, I'm already gonna simplify, we know that these three balls move together. So just like in the X case, they're all gonna have that same Y velocity. So I'm gonna factor those masses M one plus M two plus M three and multiply by the final velocity in the Y direction. OK. They're moving together as one with the same velocity. Now remember ball one and ball three were moving to the west, their entire initial velocity was in the X direction. So now that we're looking at the Y component, we know that their Y component of their velocity is zero. So the first and the third term on the left hand side go to zero and we're left with just this second ball. Now the second ball has a mass of 600 g. So we have 600 g multiplied by its velocity. It was moving to the South which is in the negative direction. So this is negative six m per second. And on the right hand side, we have the some of those three masses 400 g who lost 600 g, who lost 800 g multiplied by the final velocity in the Y direction. V F Y simplifying we get negative 3600 g meter per second. On the left hand side, this is equal to 1800 g multiplied by V F Y on the right hand side. And when we solve for V F Y, we get a value of negative two m per second. OK. So we have the X component, we have the Y component of our velocity, we can put these together to find the magnitude and direction. Now, uh the X component of our velocity is negative, that means that we're moving to the left. And so we have some X component which has a magnitude of 4.266 m per second. And we have a Y component also negative. That means pointing down with a magnitude of two m per second. So our final velocity is going to be the hypotenuse here connecting these two down into the left. We're gonna call it the F and I've written V F without the vector symbol indicating that we're talking about the magnitude and we have our angle theta here. Now we do some triangle math, we can use Python theorem to find V F V F squared is going to be equal to 4.266 repeated meters per second squared plus two m per second squared. OK. AC squared equals A squared plus B squared. Simplifying we get that V F squared is equal to point 2044 repeated meters squared per second squared. And if we take the square root, now when we take the square root, we're gonna get both the positive and negative root here, we're looking for speed. So we're just looking for the magnitude. And so we're just gonna take the positive route and that is going to be 4.71216 m per second. OK. That final speed and we were also asked for the final direction. So we want to find this value of data and theta is gonna be related to these sides through the tangent. OK? Because we're looking at the opposite and adjacent sides. So the is going to be the inverse tangent of the opposite side which is two m per second divided by the adjacent side. 4.266 repeated meters per second. And we get that theta is equal to 25.115 degrees. All right. So we found our speed, we found our direction. Now we can go back and answer this question and one thing before we move away from this diagram, OK. Theta is going from the horizontal and pointing downwards. OK? The horizontal pointing left is west and downwards is south. So this angle is south of west. OK. So looking at our answer choices and rounding to three significant digits, we found that the speed of these balls after they collide and stick together is 4.71 m per second with an angle of 25.1 degrees south of west. This corresponds with answer choice D thanks everyone for watching. I hope this video helped see you in the next one.