A 100 g ball moving to the right at 4.0 m/s collides head-on with a 200g ball that is moving to the left at 3.0 m/s. a. If the collision is perfectly elastic, what are the speed and direction of each ball after the collision?

Video transcript

Hey, everyone in this problem, we have two marbles that collide head on in an elastic collision. OK. We have marble A with a mass of 14 g that's initially moving to the right at 2.5 m per second. And marble B with the mass of 12 g moving to the left initially at 3.2 m per second. OK. So let's draw out what we were given so far and then we're gonna get back to the rest of this problem. So we have our two marbles, we have marble A which we're gonna draw in red. We're told that it has a mass of 14 g. So ma is equal to 14 g and that its initial velocity, it's to the right uh with a speed of 2.5 m per second. So we're gonna take to the right as positive. That means that that speed or that velocity story is going to be positive. It's in our positive direction. Now, we also have marble B which we're gonna draw in blue marble B has a mass M B of 12 g and it's initially moving to the left at 3.2 m per second. And so the initial velocity is going to be negative 3.2 m per second. OK. It's negative because it's moving to the left and we've chosen to the right as our positive direction. OK. So that's what we have going on here. Now, we're told to assume that the right is positive as we did in our diagram, we are asked to find the final speed and direction of each marble after the collision. And again, we're assuming this collision is perfectly elastic. Now, we're given four answer choices and they each contain a different combination of final velocities for marble A and marble B. And we are gonna come back to those as we work through this problem. Now, we have a collision problem. The first thing we want to think about is our conservation of momentum. We know that we have momentum conserved, we have no net external forces acting. So we have P not the initial momentum is equal to P F the final momentum. Now, our momentum is gonna be made up of two things. Hey, marble, A and Marble B. So we have the momentum initially of marble, A peanut A plus the momentum initially of marble B P, not B and that is equal to the momentum, final of marble A P FA plus the momentum, final of marble B P F B. OK. So any time we're using not, it indicates before the collision, anytime we're using F it indicates after the collision and then the A and B relate to marble A or Marble B. Now recall that the momentum is equal to the mass multiplied by the velocity. So for each of these terms, we're gonna have the corresponding mass multiplied by the corresponding velocity. And so we have MA V not A plus M B V, not B is equal to MA V FA plus M B V F B. All right. So a lot of things in this equation, let's substitute in all of the values we know and see what we're left with. Now, the mass of marble, a 14 grips multiplied by its initial velocity 2.5 m per second plus massive B 12 g multiplied by its initial velocity negative 3.2 m per second. OK? And it's important to include that negative when we're talking about momentum, momentum is a vector. It contains information about the direction just like the velocity. That's why we, we've used the vector symbol. OK. So we need to include that negative and this is gonna be equal to the mass of a 14 g multiplied by its final velocity V F. Hey plus the mass of B 12 g multiplied by its final velocity V B. Now we're gonna simplify on the left hand side, we have just numbers. So simplifying that down, we get negative 3.4 g meter per second. OK. On the right hand side, we can't really simplify any further, we have V FA and V F B, we're trying to find both of those, but we have two unknowns in a single equation. OK. So this equation alone is not going to allow us to solve for both of those. But let's try to write one in terms of the other. So we can at least see the relationship between the two. So we have 14 g multiplied by V fa plus 12 g, multiplied by V F B on the right hand side. Now let's isolate for V FA. In order to do that, we need to move this 12 g multiplied by V F B to the left hand side by subtracting. And then we're gonna divide by 14 g and we get that V FA is equal to negative 3.4 g meter per second minus 12 g. V F B all divided by 14 g. Well, you'll notice that we've used grams as our mass in this equation. OK? Our standard unit is kilograms. So in most cases, we convert to kilograms. In this case, we didn't need to because the unit of mass is gonna cancel up. OK. We have grams in the numerator, we have grams in the denominator. Those are all gonna divide up. OK? And so it didn't matter what unit we used for mass as long as we were consistent. OK? But you do need to be careful. In some cases, you will need to convert into your kilograms. Simplifying one more line here, negative 3.4 g meter per second, divided by 14 g. Again, that gram is gonna cancel out. We get negative 0.243 m per second K minus 12 g divided by g V F B which gives us negative 0.857 V F B. So now we've written the velocity V fa in terms of the velocity V F B. OK. So we have a relationship between the two. How can we figure out the actual value of these? OK. Well, remember we're told that this is a perfectly elastic collision. So not only do we have conservation of momentum but we also have conservation of kinetic energy. Now, we could go ahead and use our equation for kinetic energy where we have K knot is equal to K F K. The initial kinetic energy is equal to the final kinetic energy and we could solve. And that is gonna give us the correct answer. It's gonna be a little bit messy to work through because we'll have to fill in or substitute in this relationship in equation one that we've found a simpler way to do this is to recall our coefficient of restitution. OK. And that coefficient of restitution is directly related to this conservation of kinetic energy. OK. That's where this equation comes from. And it's just gonna be simpler to use. So recall that we have e that coefficient is gonna be equal to the absolute value A V fa minus V F B divided by the absolute value of V, not a minus V not. And in a perfectly elastic collision, that coefficient is equal to one. So now we know that we have one is equal to V fa. OK. Well, we can substitute in equation one for V fa we know that V fa is negative 0. m per second minus 0.857 V F B. We're gonna subtract V F B from this. So we have minus V F B and all of that is divided by the absolute value of 2.5 m. Her second minus negative 3.2 m per second. So now this is a relatively simple equation for us to solve there's some absolute values. So we just have to be careful there. But otherwise we only have one unknown V F B. We can go ahead and solve in the denominator. We're gonna get 5.7 m per second. When we simplify, let's multiply both sides by 5.7. I need to get rid of that denominator. So we have 5.7 m per second is equal to the absolute value of negative 0.243 m per second minus 0. V F B minus V F B. Now there are two cases. Yeah, we have our absolute value. So we need to consider both cases. The first case, if everything inside of the absolute value is positive, we can just drop those absolute value bars. We get 5.7 m per second is equal to negative 0.243 m per second minus 1.857 V F B. Now let's solve this side first. Before moving to the other, we're gonna add 0.243 to both sides. We get 5.943 m per second is equal to negative 1. multiplied by V F B. And we get that, that final velocity V F B is equal to negative 3.2 m per second. And what you'll notice is that this is actually the initial velocity of the marble bee. OK. So we have two possible velocities. The first one we found is that initial velocity. So this is not what we're looking for. OK. After the collision, we expect that the velocity of marble B will have changed. So let's look at the second option. That second option, if what's inside the absolute value is negative, what we do is we multiply by a negative to make it positive. OK. So what we're gonna get in this case is negative 5.7 m per second is equal to negative 0.243 m per second minus 1.857 V F B. OK. So instead of having a positive, we now have a negative on the left hand side, we've multiplied that entire expression by a negative. Now adding 0.243 to both sides is gonna give us negative 5.457 meters per second, which is equal to negative 1.857 V F B. So we get that V F B is equal to 2.94 m per second. And that makes more sense. OK? We have this smaller ball and B that marble B that's moving towards marble A, they're gonna collide, OK. That's gonna cause marble B to start moving to the right. It was initially moving left, that's gonna cause it to move to the right now in the positive direction. So it makes sense that we get this positive velocity. Remember the question wanted the final velocity for both Marvel. So we have the final velocity for Marvel beat. We've already written a relationship between marble A and Marvel B. So let's get back to equation one that we wrote down to find the final velocity of marble A. So the final velocity of marble A B fa is equal to negative 0.243 meters per second minus 0. multiplied 52.94 meters per second. And this is gonna give us a final velocity of marble A of negative 2.76 m per second. OK? And so we now have that final velocity for both marbles and we can compare this to our answer choices. So for marble A, we found the final velocity is negative 2.76 m per second. For marble B, it's 2.94 m per second. And this corresponds with answer choice. D thanks everyone for watching. I hope this video helped see you in the next one.