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say there's a basketball heading straight toward a scoop of peanut butter chocolate chip ice cream so these are going to collide there's different ways you could characterize this collision but one thing that physicists are almost always interested in is whether this collision is going to be elastic or inelastic so what does it mean to say a collision is elastic well elastic collision is one where the kinetic energy is conserved and I don't just mean the kinetic energy of one of the objects I mean the total kinetic energy of all the objects this is where the total kinetic energy of all colliding objects is conserved and don't forget people get confused about this word conserved that's really just a fancy way of saying the total amount of kinetic energy is constant ie it remains the same value before and after a collision and we can put this into a mathematical statement if we're clever we could say all right total kinetic energy conserved so if we just write down the basketball has some kinetic energy before the collision I'm just going to use the letter K for kinetic energy so I'm going to connect energy the basketball that's going to be before the collision so we need another subscript this is going to get a little messy I'm have two subscripts wanted to know which object I'm talking about the B will be for basketball and the second letter is going to represent when I'm talking about it ie this is going to represent initial like before the collision so this is the initial kinetic energy of the basketball and if we add to that because we want the total kinetic energy if we add to that the kinetic energy that the scoop of ice-cream had I'll use s for scoop of ice-cream and initially this would represent the total kinetic energy before the collision and we could do the same thing for after the collision we could say that the basketball is probably going to be moving after the collision so the basketball will have some final kinetic energy and if we add to that the kinetic energy that the scoop of ice-cream had after the collision ie finally this here would be the total kinetic energy after the collision well if the collision is elastic that means the total kinetic energy is conserved that means that this total initial kinetic energy has to equal this total final kinetic energy I could just say that these two are equal if it's an elastic collision and this is what we mean by a collision being elastic it means that the total kinetic energy is conserved for an inelastic collision the total kinetic energy is not conserved in other words this expression doesn't hold so if I put that over here if it's inelastic what you can say is that the total initial kinetic energy does not equal the total final kinetic energy and for most inelastic collisions the initial total kinetic energy is greater than the final total kinetic energy in other words in an inelastic collision you'll lose some kinetic energy some of this kinetic energy gets transformed into some other kind of energy and that energy is typically thermal energy because think about it if this if this ice cream scoop splatters right into the basketball and the atoms and molecules that make up the ice cream scoop so this ice cream scoop is made out of atoms and molecules delicious atoms and molecules and they're not masses connected to the springs but roughly speaking you can think of a solid as masses little tiny molecules or atoms connected by Springs it's really electromagnetic forces here and chemical bonds going on but that's complicated to just get a nice visual picture of what's happening imagine this collision happens well that's going to cause this atom or molecule to start oscillating more than it was this one's going to start oscillating more than it was and since these atoms and molecules now have more kinetic energy on their own this random thermal energy the total kinetic energy that this whole ice-cream scoop is going to have going forward is going to be less because some of that's going to be distributed randomly amongst the atoms and molecules in the ice cream scoop now if it's a really melted ice cream scoop if the ice cream scoop is not very cold there these Springs are not going to be very stiff these atoms and molecules can just slide around however they want there might be a lot of energy a lot of kinetic energy that it's turned into thermal energy but if you freeze this ice cream scoop if you like to take this thing straight out of the deep freezer then these bonds are going to be a lot stiffer and these atoms and molecules are going to be much more stuck in place than they were previously so once this structure becomes more rigid it's harder to transfer that kinetic energy into these individual atoms and molecules and it will become more and more elastic you'll waste less and less kinetic energy to this thermal energy here and if you take this idea to the extreme if you instead try to take a steel ball where these bonds between atoms are extremely stiff and rigid you start to approach a collision that might be considered elastic because your final kinetic energy might be almost the same as your initial kinetic energy now if I were you I might be like hold on a minute total kinetic energy is not conserved but we just said that kinetic energy in the collision goes into kinetic energy of these molecules that's still kinetic energy right thermal energy is still mostly kinetic energy and yeah it's true thermal energy is mostly kinetic energy I mean there could be a little potential energy and different kinds of energy in there as well when you're dealing with thermal energies but it is mostly kinetic energy so we should make a distinction when we say total kinetic energy is conserved we mean the total kinetic energy of that macroscopic object moving in a certain direction so the speeds in other words that we're talking about and these kinetic energies are the speeds of the macroscopic objects right of the ice cream scoop itself not of the individual atoms and molecules in other words we're not going to include the random jiggling kinetic energy that these atoms and molecules have in this calculation over here otherwise basically every collision would be elastic because yeah that macroscopic kinetic energy turns into microscopic kinetic energy but up here we're talking about the macroscopic kinetic energy of that entire object moving in a certain direction so to make this clear let's show an example with some numbers here let's just say this basketball and this scoop of ice cream had a certain speed before the collision so let's say this basketball was going 10 meters per second before the collision and the ice cream scoop was going let's say 8 meters per second and let's say after they collide this basketball still moving to the right but it's only moving at about one meter per second let's say and the scoop of ice-cream let's say it gets knocked backward and it's now going five meters per second to the right and I looked up the mass of a basketball the mass of a basketball is about 0.65 kilograms and now with that mass of the basketball I have to pick the right mass over here for my mass of the ice-cream because I picked these velocities just kind of randomly so in order to conserve momentum for this collision and almost all collisions should be conserving momentum the mass of this scoop of ice-cream should be about 0.45 kilograms now with these numbers in here we can ask was this collision elastic or inelastic and one mistake people make is they say oh well they bounced off of each other right because this basketball is going to the right at only one meter per second and the scoop of ice cream is going to the right at five meters per second you must have bounced off of each other they separated doesn't that mean elastic and no that doesn't mean elastic just because they bounce just because they bounce off of each other does not imply that it's elastic it works the other way if it's elastic they do have to bounce off of each other but just because it bounces does not mean it's elastic so be careful there just because they bounce here does not mean it's elastic what do we do to check whether it's elastic what we do is we check whether the total kinetic energy was conserved or not so let's just check we've got enough numbers here to figure that out so I can use the formula for kinetic energy which is one-half MV squared and I can find what is the initial kinetic energy of the basketball it'd be one-half mass of the basketball times the initial speed of the basketball which was ten so I'm using initial speeds here because I want to find the initial kinetic energy and then I have to add to that because I want the total kinetic energy I have to add to that the initial kinetic energy of the scoop of ice cream so it's going to be plus another one-half times the mass of the scoop of ice cream times its initial speed which was eight meters per second you might say isn't it negative eight well we're going to square this anyway so it doesn't matter so don't forget the square and if we add all those up we get forty six point nine Jools of total initial kinetic energy so is this equal to the final amount let's just find out the final amount of kinetic energy if I take the final speed of the basketball and use that to find the final kinetic energy of the basketball I'd have one-half mass of the basketball times the final speed is only one meter per second and I still square it and then I have to add to that the final kinetic energy of the scoop of ice cream which is going to be one-half the mass of the scoop of ice cream times five squared because five was the final speed of the scoop of ice cream and if I add all that up I get that this equals five point nine five joules of total final kinetic energy so is this collision elastic no way is not even close this initial total kinetic energy was forty six point nine joules this final total kinetic energy was five point nine five joules the kinetic energy here was not conserved and because it was not conserved we would consider this and in elastic collision but if you're clever you can just look at the numbers here you didn't actually have to go through all this work you could just say hey the basketball started with ten meters per second it ends with one meter per second it's definitely got less kinetic energy than it did before and this ice cream scoop started with eight meters per second and it ends with five meters per second it also ends with less kinetic energy than it did before so this final kinetic energy has to be smaller than the total initial kinetic energy and you can ask where did that energy go it goes into the thermal energy of these molecules and atoms in the object's vibrating thermally a little more than they did before including in the basketball as well as sound waves that can get created that also takes away energy there's lots of places for energy leaks and in this particular collision there were a lot of leaks because we lost a good majority of the kinetic energy that we started with which made this an inelastic collision so recapping for a collision to be elastic it's not enough to just know it bounces you have to see if the total initial kinetic energy is the same as the total final kinetic energy if that's the case it's an elastic collision and if that's not the case it's an inelastic collision one last note sometimes you'll hear the word perfectly elastic collision well that's redundant that's just another way to say an elastic collision in other words a collision where the initial kinetic energy really is equal to the final kinetic energy but you'll also sometimes hear about a perfectly inelastic collision and this is meaningful this means that the two objects that collide stick together so if it's perfectly inelastic this means that they must stick together and move off as a single unit in other words if this scoop of ice cream splattered into the basketball and then stuck to it and the two moved off to the right at some speed that would be a perfectly inelastic collision now whether it's elastic or inelastic momentum is still going to be conserved for these collisions if that collision happens over a short time interval there's not enough time for an external force to cause enough impulse to impact the momentum greatly so if it's one of these instantaneous impacts that happen in collisions then the momentum will be conserved for both elastic collisions and inelastic collisions sometimes people get confused they're like wait I know that energy is only conserved for elastic collisions maybe that means that momentum is only conserved for elastic collisions but that's not true momentum will be conserved for both inelastic and elastic collisions you might object you might be like wait wait wait if you're clever you might be like hold on in these inelastic collisions we're losing all kinds of energy to the random thermal oscillations in this material aren't we also losing momentum to those random oscillations I mean movement implies both kinetic energy and momentum so why aren't we losing momentum in these inelastic collisions and the reason is the oscillations of the atoms and molecules in this material but they're oscillating randomly in random directions this thermal energy gets distributed in a random way so that the momentum of the atoms and molecules in that structure cancel out because if you've got momentum in every single direction and momentum as a vector that equals no momentum at least no net momentum because these are all going to cancel out this one cancels with this one this one cancels with that one that one cancels with that one so that's why in an inelastic collision there's no loss of total momentum to the microscopic atoms and molecules of the object but there is a loss of kinetic energy because kinetic energy is a scalar kinetic energy has no direction kinetic energy can't cancel in this way because it's not a vector so even though in an inelastic collision you lose kinetic energy to the microscopic atoms and molecules you don't lose any momentum to them because all that momentum just cancels out and the bulk motion of these macroscopic objects must maintain the total momentum and this is wonderful news actually because that means momentum is going to be conserved for both elastic and inelastic collisions doesn't matter what kind of collision it is momentum is going to be conserved as long as there's no time for any net external impulse to act during that collision so even though energy is only conserved for elastic collisions momentum will be conserved for every collision

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