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Directions: Questions 1, 2, and 3 are short free-response questions that require about 13 minutes to answer and are worth 8 points. Questions 4 and 5 are long free-response questions that require about 25 minutes each to answer and are worth 13 points each. Show your work for each part in the space provided after that part.A car of known mass $m_{1}$ will collide with a second car of known mass $m_{2}$. The collision will be head on, and both cars will only move linearly both before and after the collision. In a clear, coherent, paragraph-length response, explain a method for determining whether the collision is perfectly elastic, perfectly inelastic, or neither. If the collision is perfectly inelastic, include at least one possible cause of energy loss.

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in this example, we are going to look at collisions and review some of the things that are true about collisions. Um and we'll start off with two cars that are going to run into each other with a one dimensional head on collision and we'll presume that we know the mass of each car and the initial velocity vector along the line for each of those. And they're headed towards some head on type situation. Sorry, that looks a little bit more like a flying saucer, but you get the idea it's a car. Yeah. And there interacting on a straight line. So one dimension just to make things easy. Um and the first thing to realize is that as long as the external forces are negligible, they don't have to be identically zero. But uh very small forces such as friction, much smaller than the collision forces that these interact with. Um what is true is the sum of the momentum, which is mvs initial have to equal the sum of momenta afterwards after the collision. And that's known as momentum conservation. We could actually write a formula for it. In this particular example, we would say that M1 if you want initial minus M. Two uh B two initial equals and then we'd have to have some picture of the collision afterwards. So let me kind of show a picture after the collision. So we can have a little bit more clear picture of what's going on. Let's pose that the first car continues after the collision. To the right with some new velocity fee to final Be one final, sorry. And our second car gets bumped backwards. Um Okay, that looks a little bit more like ufo again, but be too final. Our momentum conservation then would have both of those momentum going in the same direction to the right, which I'm calling positive in this case. Um And so we would have momentum conserved. Um So that's true for any collision, that you're going to let momentum conservation hold true. Uh It's a little bit different and how collisions get classified from there on it has to do with energy conservation. And in particular kinetic energy, a perfectly elastic collision is one in which the some of the kinetic energies beforehand totally add up to the kinetic energies afterwards. So that's known as a perfectly elastic collision. And the sum of kinetic energies before identically add up to the sum of kinetic energies afterwards. And we could write another formula for our little example here, We'd have 1/2 and one V one initial squared plus one half. I am too. The two initial squared is equal to the same sort of some but with the final velocities uh huh. After the collision coming into play, I was and I'm just about out of space there. Uh Let me see if I can get a little bit more space. There we go. Okay, so we could write down a relationship for that. And if that relationship holds then you have a perfectly elastic collision. A perfectly inelastic collision is one in which you're absolutely guaranteed that a lot of energy in the form of kinetic energy will get lost two other forms. So the kinetic energy afterwards is going to be much, much less than this. Total kinetic energy before the collision. Yeah, so energy is never created nor destroyed. It just converts from one form to another. Um But what's also true about perfectly elastic is not only is kinetic energy lost, but you also have the condition that the cars stick together. And I can say that simply their final velocities have to agree with each other. They're traveling together afterwards, uh stuck together in some way, shape or form. Um Whether there could get all tangled up in each other um Could happen with cars. Uh huh. But with people they could just be holding on to each other. And the question is, where does the energy go? Um How does the energy convert? Uh can again and again and g. That the system had beforehand can get into converted into a number of forms. And we'll just write down some possibility possibilities. One is in the form of heat and there are several ideas for how the heat could happen on his friction. Mhm. Another idea is that the atoms and molecules, let's just say Adams to make life easy in the cars can start to var vibrate and eventually those vibrations die down. Um There's all sorts of ways for that vibrational energy can get transferred into heat. Um Or sound even and that's another way that sound, the sound could come out of the collision and that's another way for the energy to get lost. Um There's also the understanding of the atomic bonds getting broken or actually absorbing some of the energy. Just like if you had to compress the spring um you have to put energy into that spring to make it compress. So atomic bonds are kind of like springs and they can absorb energy or they can break. And that of course takes energy uh to transform atomic bonds. Mhm. Okay so that's different between perfectly and elastic uh and perfectly elastic and sometimes it's good just to think um Kind of phenomena logically perfectly an elastic collisions are what you might call sticky things stick together whereas perfectly elastic are bouncy. Yeah the objects have a tendency to bounce off of each other but that's not a very good quantitative discussion

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