Elastic Collisions and Stored Energy

this car is gaining momentum and kinetic energy and now there's no longer any momentum in kinetic energy the Collision was very complex involving the cars and the Earth here in slow motion is a simpler Collision but it's still complicated we'd like to look at two bodies which are not interchanging momentum and kinetic energy with the Earth to do this we must really isolate the bodies involved in the Collision in this way we'll better understand what kind of interaction is occurring now I've done the best that I can let's look at some collisions in this isolated system where there are no forces to transfer momentum and kinetic energy to the table and thus to the earth almost all the kinetic energy disappeared it went into heating up this blob of putty now let's look at another some of the kinetic energy disappeared but you see them move away so not all of it disappear now I've got two other pucks over here I'll get these off the table there's a magnet in each one of these as you can see and if I push one of these pucks it slows down and almost stops and the other Puck moves off with a velocity equal to the velocity this one had originally the kinetic energy that this Puck lost was all transferred to this one in this Collision the kinetic energy after the Collision was equal to the kinetic energy before we call this kind of collision elastic in the other two collisions we saw kinetic energy was somehow spilled or lost during the transfer from one body to another those collisions we call inelastic but in all these collisions whether kinetic energy is lost or not the total momentum of these pucks Remains the Same now you already know about the conservation of momentum and the transfer of momentum from one body to another this film is about the transfer of kinetic energy from one body to another now today we're going to concentrate on elastic collisions in an isolated system we'll analyze these collisions in detail and in analyzing them we'll learn much more about how kinetic energy is transferred from one body to another these collisions happen slowly enough so that we can observe the kinetic energy in detail not only before and after the Collision but during the actual transfer of kinetic energy indeed the conclusions we draw about the transfer of kinetic energy will lead us to another important physical concept I'll compute the kinetic energy 1 12 mv^ squ of each of these pucks at various times during the Collision I filled these pucks with dry ice so they each have the same mass now I've mounted a camera directly above the center of the table I'll use it to take stroboscopic pictures of these collisions the camera will record the position of these pucks at regular intervals of time the bullseye here marks the center of the puck now let's try some Collision now watch it from directly overhead let's look at that one again and this time you'll see the strobe flash this is the stroboscopic picture of the collision at equal time intervals the flashes recorded the position of the bullseye for each Puck giving us tracks like this let's Analyze This the average velocities of the pucks here and here were continually changing that is the distance each Puck moves between successive flashes with is changing I want to be able to calculate the average velocity of each Puck and for convenience I'll measure the distance traveled by each puck in the time interval between successive two flashes over here at the first Flash the puck was here then two flashes later it was here I'll call this interval number one then two flashes later the puck had moved to here interval number two three 18 and 19 now off hand I don't know just during which time interval this second Puck started to move but I do know that at the last Flash the first Puck was here and the second here so this then is the 19th interval for the second Puck and the 18th and the 17th the E and the seventh one is right here now we can't make any measurements in here but obviously the puck was moving probably in both the fifth and the sixth interval all right let's make some measurements I'm going to make my measurements in millimeters and measure to the closest half millim now in the first interval it move from here to here 28.0 and in the second interval 27.5 third interval 27.5 fairly constant 27.0 fifth interval 27.0 and in the sixth time interval 25.5 and this Puck has probably started to move but we can't measure it I'll put a question mark in the column 23 and in this time interval the second Puck has moved a measurable amount that's 5 mm 23 5.0 19 and 10 and 22.5 those last several readings were quite constant so that's as far as we'll make measurements now I have recorded all the distances moved by the pucks during the various time intervals and now I must compute the kinetic energy for each puck EK is equal to 12 mv^ 2 and the average velocity during each interval Delta D / delta T the velocity squared is Delta d^2 ID Delta t^2 now the kinetic energy 12 M Delta d^2 / Delta t^2 the mass and the time interval is constant throughout the Collision so all of this is a constant and we can write the kinetic energy EK K is equal to a constant times Delta D2 Delta D is just what we have obtained from the photograph if I Square these numbers I can use them as a measure of the kinetic energy Delta d^2 is proportional to the kinetic energy and a variation of Delta d^2 will be the same relative variation as the kinetic energy I'll use this relative measurement of energy in all my calculations now add a column for the kinetic energy of each Puck ek1 ek2 here I've calculated the kinetic energy for each of the two pucks and the total kinetic energy of both pucks the kinetic energy of the first Puck changed slowly here because the puck wobbled slightly just after I pushed it and then the kinetic energy really began to decrease still later it was constant again but less than before just at the time the kinetic energy of the first Puck was really beginning to change the second Puck began to gain some kinetic energy right here at about the fifth or sixth time interval we can say the Collision started second Puck gained in kinetic energy here to about here and then it was constant right here at the 12th or 13th time interval is when we can say the Collision was done the kinetic energy of both pucks before the Collision was constant and after the Collision was constant here where I tabulated the total kinetic energy of both pucks you see that kinetic energy after the Collision was the same as before so by only looking at the kinetic energy before and after the Collision we see that it is elastic however during the Collision the kinetic energy is less even though it was all there after the Collision to get a better look at what's Happening Here I've plotted the data on a graph the total kinetic energy versus time shows us that indeed during the Collision some of the kinetic energy had disappeared but after the Collision the total kinetic energy is the same as before so that this energy disappearance here was only temporary so what's going on here let's take another look at that photograph at this interval number seven at the beginning of the Collision equal forces acted on both pucks the force acting on the first Puck acted over this range this distance while a force of equal magnitude acted in this direction on the second Puck over this range a smaller distance so more kinetic energy was transferred away from the first Puck than was transferred to the second does this mean that the whole system was losing kinetic energy during this interval yes kinetic energy was being lost but this loss was only temporary since as we saw the pucks regained that kinetic energy by the time the Collision was over now we just saw that here at the beginning of the Collision the first Puck lost more kinetic energy than was gained by the second thus the net decrease in kinetic energy here at the bottom of the dip whatever kinetic energy was lost by the first Puck was gained by the second and over here near the end of the Collision where the kinetic energy was increasing both pucks were moving apart and the component of the force acting on each Puck pushed that puck transferring kinetic energy to the puck thus the net increase in kinetic energy here now this decrease of kinetic energy at the beginning of the Collision as the pucks came together and the increase as they moved apart suggest that perhaps the separation of the two pucks had something to do with the amount of kinetic energy lost here to check this let's measure the average separation of the two pucks during the Collision the distance between the two pucks was continually changing so an average distance between the pucks would be a measurement between the center of each time interval for instance here in the eth interval the average distance would be from here to here all right in the first time interval the average separation is 28.5 millimeters and in the second 160 180 mm and the last is 228.50 now just by looking at this data we can see that as the pucks got closer together the energy decreased as the puck moved farther apart the kinetic energy increased and here where the pucks were closest together the total kinetic energy was the least once again I plotted this data here is the graphical representation of the total kinetic energy of the system versus the distance between the pucks these dots Mark the total kinetic energy of the pucks as they move together and and the circles the total kinetic energy as they move apart the graph is the same whether the pucks are coming together or moving apart so the total kinetic energy is dependent on only the distance between the pucks we can say that another way out here where the pucks are far apart the total kinetic energy is this much this is just the amount of kinetic energy I gave the first Puck when I pushed it as the pucks move closer together some of the kinetic energy disappears it is transferred into storage this transfer of kinetic energy is caused by the forces acting on the pucks transferring kinetic energy into storage as the pucks move closer together and transferring it out of storage as the pucks move farther apart to make this clearer I'm going to plot the stored energy versus the distance between the pucks for instance when the pucks are this far apart the amount of energy stored is this or when the pucks are at this distance the stored energy is that and here is the graph of stored energy versus distance between the pucks what this means is that the amount of energy stored when the pucks are just a distance s apart is the same whether the pucks are coming together or whether they're moving apart now we've only looked at one Collision does this relationship hold for all collisions of these pucks let's look at some other collisions here's an almost head-on collision of the magnet pucks now I push the puck faster this time you see the strobe light flashing over a more glancing Collision this is the stroboscopic photograph of that last Collision we get the same kind of record of the position of these two Pucks in this Collision as we got in the last Collision I've already analyzed this collision and computed the total kinetic energy of these pucks versus their distance apart and found from that the energy stored by these two Pucks in this Collision versus their distance apart you remember in the first Collision the curve of stored energy versus distance between the pucks now for this Collision I've plotted that same data on a plastic overlay and the coordinate system on this overlay is identical to that on the graph the data for this Collision of the magnet pucks lies right along this curve indeed for all collisions of these two pucks that I've analyzed I always get the same curve in some cases the pucks got closer together than in others this just means the curve only went this far or was extended out Beyond this point but no matter how I aimed the pucks or how much initial kinetic energy I gave them I always got this same unique function of stored energy versus distance let's take a closer look at the stored energy it's stored in such a way that it's later regained as kinetic energy for instance if the pucks are a specific distance apart let's say this distance then they have stored up this amount of energy now it doesn't matter how slow or fast this Collision takes place there is always a specific amount of energy stored when the pucks are just this distance apart for instance during the Collision when the pucks were right here we could hook them together and then come back tomorrow or in a week and the energy would still be stored there they would still have that ability that potential to acquire kinetic energy when they were finally unhooked and could move apart again let me show you what I mean if I push these two pucks together and then release them they fly apart and I could determine their kinetic energy now I've got string here that's pretty cold this will hold the puck that same specific distance apart that I just indicated on the graph we know that when they're just this distance apart there's a unique amount of energy stored by these two pucks which is later transferred to the kinetic energy of the as they fly apart okay let's burn the string there they go flying apart here's the explosion seen from above and this time there are the strobe flashes this is the stroboscopic photo photograph of that explosion and I've recorded the same kind of data for the position of these Pucks in this explosion as I have for the other collisions these white streaks here are just the images of the burning fragments of string or the match now before the explosion the pucks were just 56 12 mm M apart at that distance they had no kinetic energy then I burned the strength and the pucks moved apart the forces acting on them transferred kinetic energy to them this kinetic energy that was transferred to the pucks had been stored in the system of the two pucks when they were just this far apart we predicted that that amount of stored energy was just this much when the pups were this far apart now I've computed the total kinetic energy of the pucks after they exploded and plotted this on a plastic overlay so that we could compare it to the energy store here when the pups were tied together with a string they were 562 mm apart then as they exploded they moved farther apart and the total kinetic energy they had they acquired was this much and it is the same as the energy that we predicted would be stored by the two pucks when they were just this distance apart so it doesn't matter how we get the pucks to be just this far apart whether we hold the pucks together and let them fly apart or whether we push one Puck into the other the amount of energy stored when the pucks are a specific distance apart is always the same this kind of recoverable stored energy we call potential energy now we've computed the potential energy here by measuring the amount of kinetic energy which disappears there are other ways of doing this if you knew how the force acting on the pucks varied with distance you could compute the work the energy transferred to the Pucks in pushing the pucks from way out here where the force is negligible into here the energy transferred to the pucks by pushing them into this distance apart is just equal to the potential energy stored by the P when they are just this distance apart anytime that kinetic energy can be transferred into storage and then back out again the Collision is an elastic one we've examined several elastic collisions and in each case we have found a unique relationship between the stored energy and the distance between the pucks and whenever we have this this unique storage function we have a potential energy it is our definition of potential energy now in these experiments we have also seen that whenever the kinetic energy decreases by a certain amount the potential energy increases by that same amount so we can say something else about elastic Collision that during the Collision the sum of the kinetic and potential energy remains the same it is conserved the sum of the kinetic and potential energy of a group of bodies is called its mechanical energy in an elastic Collision we always get a unique energy storage function and mechanical energy is conserved we find a potential energy associated with many bodies other than these pucks there is a gravitational potential energy associated with any two masses and an electrical potential energy associated with charged particles here we've only discussed potential energy in an isolated system but in any interaction whether we can isolate it or not if there is a unique function of stored energy versus the separation of the bodies then this concept of potential energy will be useful for