Welcome to module 12 of two dimensional dynamics. Our learning outcomes are to define the Coefficient of Res, Restitution and give examples, and to solve an Impact problem. So we left off last time where we had solved a conservation of momentum problem for two blocks that stuck and moved off together. And so let's go back over and look at this situation. We had one block coming in with a velocity of v when they moved off together. If they stuck together, their total velocity for both blocks after the collision was v over two. But I said last time that in most collisions we have some rebound if you will, and so they don't move off at necessarily the same velocity. And in that case we have to have another piece of information which we're going to call the Coefficient of Restitution. And here is the definition of the Coefficient of Restitution. It is that the relative velocity of the bodies after impact is divided by the relative velocity of the bodies before impact. You can find the development in any standard dynamics text, but this is what it ends up coming out to be. You'll notice that the B's and the A's are reversed but you can see that in the derivation when you when, when you research that. [COUGH] so we can have a range of values for the Coefficient of Restitution. If e is equal to 0, we call that a perfectly plastic collision. And if e is equal to 1, we call that a perfectly elastic collision. For a perfectly plastic collision, let's give it an example. Let's say that body A is the earth, or the ground if you will, and we're going to give it a velocity of 0. Body B is my other body. And so if I have a body coming in with a certain velocity, v B1, and after it smacks the other body, the Earth in this case. If it has zero velocity then it's called a perfectly plastic collision, and e is equal to 0. And so, if we slide back over here, that was actually the case we had with the two blocks because they moved together. They didn't separate. And so the velocity was the same after the collision. Here's another example, I have silly playdoh ball here. If it comes down and drops and hits the ground and does not bounce back up, then that would be a perfectly plastic collision. Now in reality, this does bounce up a little bit so it does have a little bit of a Coefficient of Restitution. Let's go back over here. If the velocity after the collision is equal to the, the relative velocity after the input, is equal to the relative velocity before the impact, then we have a Coefficient of Restitution of 1. So, in that case, let's go back over here. Here I have a superball. If the velocity before impact is equal to the velocity after impact, and we call that a, a, a perfectly elastic collision and e is equal to 1, so the ball would come back to ideally the ball would come back to its the original height. We know that's not physically possible. But you can see that this has a rather high Coefficient of Restitution, maybe around 90%. Here are some other examples, this ball maybe has a Coefficient of Restitution of less than 0.5. You saw on the intro video that I'm a golfer. And so here's a golf ball. Its Coefficient of Restitution is somewhere around 0.8. I think the regulation is that a golf ball cannot have a Coefficient of Restitution higher than 0.83. Okay so that's a feel for the Coefficient of Restitution. Let's look at impact. One type of impact we call Direct Central Impact. First of all, we define the line of action. That's a common normal to the perpendicular common normal or perpendicular to the impacting surfaces. Here's the impacting surface. A common normal is called the line of action. We have central impact if both mass centers lie on the line of action. And we call it a direct impact if both velocities, as I've shown here, also lie in the line of action. We may also have Oblique Central Impact. Central being the mass centers continuing to lie on the line of action, but one or both of the velocities do not have to lie on the line of action. And we'll see that in a worksheet that we're going to do on the next slide. One assumption, or a couple assumptions that we're going to make, are that we have smooth non-spinning bodies, and that the velocity components perpendicular to the line of action are unchanged. So we're only going to get changes in the velocity along the line of action. So here's our, our worksheet. We've got two identical hockey pucks. They're going to collide. There's a Coefficient of Restitution of 0.8 and we want to find the velocity of the, the, the pucks after collision. And so first of all, we can apply Conservation of Momentum, Linear Momentum. I'd like you to do that then come on back, write the, the equation for the Conservation of Linear Momentum. So this is what you should've come up with. Since these are identical pucks, we're going to let m A equal m B equal m. And so there's the information that we have. That's not going to be enough to solve the equation alone. But let's get started and at least look at it in the i direction. So in the i direction, I have velocity of block A, so I've got m times the velocity of block A. The i direction is the sine component of 4 feet per second. So that's going to be times 4 sine of 30 degrees. And then I've got plus m B initial, m B initial is going to be the fourth, fifth component of the 5 feet per second. So it's going to be 4 5ths, and the 5 feet per second is in the negative i direction. So that's going to be minus 5 equals m times v A final in the i direction, and I don't know that. That's what I'm trying to solve for. Plus m times v B. This is final, in the i direction. I'm sorry. That's correct. And v B final in the i direction. I can cancel the i's and I can simplify the equation so that I get v A final in the i direction plus v B final in the i direction equals minus 2. And we'll call that equation 1. We're going to also use the piece of information of the Coefficient of Restitution, and so I've got 0.8 equals v B final minus v A final, both unknowns, what we're trying to find in the problem. v A initial was 4 sine of 30 in the i direction. And v B initial was minus a 4 5ths, and v B was in the minus i direction. And if I simplify that, I get v B final in the i direction minus v A final in the i direction equals 4.8. We'll call that equation 2. I now have two equations, two unknowns. And so I can solve them by combining these equations. And so when I combine them, let's take this equation up here. 1 we've got v A final the i direction, plus I'll substitute in for v B final from equation 2, so I've got 4.8 plus v A final in the i direction equals 0. Oh, that doesn't equal 0. I'm sorry, it equals minus 0.2, right? Up here. And so I get v A final in the i direction as a vector equals minus 3.4 i. I can then substitute that result back into this equation down here to find v B final in the i direction. So I've got v B final in the i direction equals 4.8 plus v A final which is minus 3.4. Or v B final in the i direction equals 1.4 i. Okay, so we're good as far as the i direction is concerned. We now need to do the j direction components. I want you to recall that I said that the, we're assuming that the, the perpendicular velocity components to the line of action are unchanged. So the velocity component changed in the line of impact direction, but not in the the off the line of impact direction, or the j direction in this case. And so I have, in the j direction, v A initial the j direction, equals v A final in the j direction. Or the j component is the cosine of 30 degree component for body A, so I've got 4 times cosine of 30 degrees, or that is 3.46. So v A final as a vector is equal to 3.46 j. I can do v B initial now, equal to v B final. So I've got v B initial in the j direction equals v B final in the j direction. Or, the v component, j component is the 3 5ths component along this slope. So I've got 3 5ths, it's in the minus j direction. So, it's going to be times minus 5 or minus 3. So v B final in the j direction, initiative in the j direction as well, is equal to minus 3 j. I put all those results together. I have the total velocity of the pucks, after the collision. And so I've solved the problem. And that's a good stopping point for this module.
