A 5.5-kg object moving in the +𝓍 direction at 6.5 m/s collide...

Question

A 5.5-kg object moving in the +𝓍 direction at 6.5 m/s collides head-on with an 8.0-kg object moving in the ―𝓍 direction at 4.0 m/s. Determine the final velocity of each object if:

(b) the collision is elastic;

Accepted Answer

Hey, everyone in this problem, two hockey pucks slide on a frictionless ice surface towards each other. The first puck with a mass of 0.15 kg moved to the right at a speed of 5 m per second. While the second puck with a mass of 0.2 kg moves to the left at 3 m per second. Assuming the collision between the pucks is perfectly elastic. We're asked to calculate the final velocities of both pucks after the collision. And we're told to consider motion to the right to be in the positive direction. We have four answer choices a through D. In this case, each containing a different final velocity for those two pucks, we're gonna come back to those answer choices once we're done working through the problem, now, we're gonna start with a little diagram of what's going on before the collision. So before the collision, we've been told that we have the first puck with a mass, we'll call m one of 0.15 kg and it is moving to the right with a speed of 5 m per second. Now, because it's moving to the right and we've been told that the right word direction is the positive direction. This has a positive velocity. So we not one, the initial velocity of puck one is gonna be equal to 5 m per second. And then we have our second puck and our second puck has a mass which we'll call M two of 0.2 kg. And it is moving to the left because it's moving to the left, it's gonna have a negative velocity. So V not two, the initial velocity of puck two is going to be negative 3 m per second. OK. So this is what we have going on before the collision. We want to figure out the velocities of both pucks after the collision. Now, this is a perfectly elastic collision because we have a perfectly elastic collision. Recall that both kinetic energy and momentum are conserved. And so what we can do is go through our conservation of kinetic energy equation and our conservation of momentum equation, we would have two unknowns and those two velocities we're looking for with two equations and we'd be able to solve. OK. But we recall that we have something that makes this a little bit easier in these cases. And that is the coefficient of restitution. Recall that this coefficient of restitution E is equal to VF one, the final velocity of puck one minus VF two, the final velocity of po two divided by would be not to minus V, not one. OK. So final velocity of P two minus final velocity of puck one. Now because we have a perfectly elastic collision, we know that E is equal to exactly one. So we know that one is gonna be equal to the difference in these final velocities. VF one minus VF two. And we know these initial velocities so we can substitute those in. So we're dividing by negative 3 m per second minus five meters per second. All right. So what you can see is now we have this equation that has these two unknowns. So it's the exact same as if we had have worked with our kinetic energy conservation equation. But it's just much simpler and there's a lot less simplifying. There's a lot fewer terms that we had to deal with a much, much simpler equation. So minus 3 m per second minus five, that's gonna give us negative 8 m per second. And we can multiply that to both sides, we get that negative 8 m per second. It's gonna be equal to VF one, mine is VF two and let's write one of these velocities in terms of the other. So we're gonna write that VF one can be written as VF two minus 8 m per second. And we're gonna call this equation one because we're gonna come back to that in just a minute. Now, we can't do anything further here. We don't know either of these values. We have one or VF two we can't solve. But we do know that we also have the conservation of momentum. So what we can do is use conservation of momentum to get another equation with these unknown values. OK. So this coefficient of restitution just simplified the calculation that we would typically do with the conservation of kinetic energy. All right. So our conservation of momentum recall tells us that the initial momentum P not is gonna be equal to the final momentum P. When we're thinking about momentum here, we have to consider the momentum from both packs. So we have the momentum initially from P one that the momentum initially from P two, that's gonna be equal to the final momentum of Puck one. What's the final momentum of Puck two recall that momentum is given by a mass multiplied by velocity. So for each of these four terms, we have that corresponding mass and corresponding velocity. This is gonna give us M one, the knock one plus M two V not two is equal to M one. VF one plus M two. VF two. Let's go ahead and substitute in the values that we know. So we know the masses of our Tupac and we know those initial velocities. So we're gonna have 0.15 kg multiplied by 5 m per second. Who lost 0.2 kg multiplied by negative 3 m per second is equal to 0.15 kg multiplied by VF one loss 0.2 kg multiplied by VF two. You can go ahead and simplify on the left hand side, this gives us 0.15 kg meters per second. And this is equal to that right hand side, 0.15 kg multiplied by VF one plus 0.2 kg multiplied by VF two. Now again, we have an equation with two unknowns. VF one and VF two. What we can do is recognize that this VF one we have in our momentum equation is the same VF one that we wrote in equation one. So we can use substitution here substitute in this expression from equation one for VF one that's gonna be written in terms of VF two. So the only unknown in our equation will be VF two, that'll allow us to solve our VF two. And once we know VF two, we can come back to equation one to solve our VF one. OK. So let's do that. We have 0.15 kilogram meters per second is equal to 0.15 kg multiplied by VF one which is gonna be VF two minus 8 m per second. Then we're adding 0.2 kg multiplied by VF two. So again, we now have an equation only in VF two. So we can go ahead and solve. Let's expand on the right hand side, we get 0.15 kilogram meters per second is equal to negative 1.2 kilogram meters per second, who was 0.15 kilograms multiplied by B two F plus 0.2 kg multiplied by V two F oops. And they should be VF two just written on the other way around there. These are the final velocity of Puck. OK. So we can continue simplifying. We're gonna move our 1.2 kg meters per second to the left hand side. We're gonna combine our two VF two terms. And what we get is that 1.35 kilogram meters per second is equal to 0.35 kg, multiplied by VF two. Last step to get VF two is to divide both sides by 0.35 kg. And so we get that VF two is equal to approximately 3.857 meters per second. OK. So this is the final velocity of Puck two. You'll notice that it's positive and this makes sense. Hey, let's go back to our picture. Initially, we have P one and P two moving towards each other. The po two is moving to the left when they collide, they're gonna bounce off each other. And because we have a positive final velocity for Puck two, that means that it's now moving to the right. OK. So that makes sense. We're gonna collide bounce off each other and now Puck two is gonna move to the right back towards where it came. That means that Puck one, we would expect to be moving to the left. We expect that that's gonna have a negative velocity and we're gonna calculate that now. So we have the F two. Now we can go back to equation one substitute in what we found to get the F one. So we have that VF one is now going to be equal to this VF two value. We just found 3.857 m per second minus 8 m per second. And this gives us a VF one value of negative 4.143 m per second. And this is approximately have a little bit of round off over there. All right. So that's the final velocity of puck one. Again, we expected it to be negative. So that makes sense. Now, we have those final velocities of both pucks after the collision, we can go back to our answer choices and wrap this up. We are gonna round to one decimal place and when we do that, we can see that the correct answer is going to be option DV one, the final velocity of puck one is negative 4.1 m per second and V two, the final velocity of puck two is 3.9 m per second. Thanks everyone for watching. I hope this video helped you in the next one.

A 5.5-kg object moving in the +𝓍 direction at 6.5 m/s collides head-on with an 8.0-kg object moving in the ―𝓍 direction at 4.0 m/s. Determine the final velocity of each object if:
(b) the collision is elastic;

Key Concepts

Conservation of Momentum

Elastic Collision

Kinetic Energy

Watch next