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Current time:0:00Total duration:12:22

Deriving the shortcut to solve elastic collision problems

Video transcript

so if you powered through the last video you saw that these elastic collision problems can get pretty nasty the algebra gets pretty ugly what we did if you missed it maybe you skipped right to this easy one and that's cool with me but what you missed or what you saw if you did watch it is that we use conservation of momentum but we had two unknown final velocities we didn't know the velocity of either object after the collision so we had to solve this expression for one of the velocities and then plug that into conservation of kinetic energy which we can do because kinetic energy is conserved for an elastic collision but since we square this expression it gets big and ugly and it gets multiplied by other stuff and now you have to combine terms and you end up just having to pray that you didn't make an algebra error or lose a sign in here and even if you do get lucky and the whole thing works out and you get an answer if you round it at all during this process your answer is going to be off by a little bit so the obvious question is is there a simpler way of solving these elastic collision problems where you've got two unknown final velocities and there is a simpler way and that's what I'm going to show you in this video so to derive this simpler way of solving the problem we're going to do what we always do in physics to derive a nice result we're going to instead of solving this problem numerically with numbers we're gonna solve it symbolically with symbols what I mean by that is instead of calling the mass of the golf ball point oh four or five kilograms let's just say this is any particular mass and we'll just call it mg for mass of the golf ball and you might be like well how's that going to help we're going to a bunch of variables in here instead of numbers it's still going to be a mess still going to get a little messy but what this does when you solve problems symbolically it oftentimes allows you to see patterns symmetries cancellations that are happening here in your calculation that aren't so obvious when there's a bunch of numbers around when it's a bunch of numbers it just looks like a big mess when you've got symbolic expressions in here sometimes something magical happens and that's what's going to happen here and it's going to give us a result that's way simpler so let's do this let's solve this problem symbolically we're going to get rid of all these numbers and we're going to turn these variables instead of calling it 40 meters per second for the initial velocity of the tennis ball we're going to leave it symbolic so instead of giving a number we're going to call this I'll call it V T for the tennis ball and then I'll write I for initial so this is the initial velocity of the this fall and we'll do the same thing for the golf ball instead of calling it 50 meters per second to the left we'll call it vgi the initial velocity of the golf ball so when I wrote 50 before I met the size of the velocity but this time when I write VG I mean the velocity in other words this VG I might be negative in fact if the golf ball is going leftward it's going to be a negative number but that's okay it's symbolic we're going to treat this V GI as the velocity so it could be a negative number or it could be a positive number if the golf ball were going to the right we're going to do the same thing for the masses we already wrote the mass of the golf balls mg now we're going to write the mass of the tennis ball as MT and now we're going to solve the same way we did before we're going to use conservation of momentum so our initial momentum would be mass of the tennis ball times the initial velocity of the tennis ball vti and then plus mass of the golf ball times the initial velocity of the golf ball v GI and you might be like wait this should be a negative sign right no I know this golf ball is going left but I'm letting this v GI be the velocity so there's a hidden negative sign in here if it's going leftward this v GI would equal some negative number in other words so I wouldn't want to put another negative or I'd be canceling off the negative that would be in here that's why I get to write a plus here and this is going to equal the final momentum of the tennis ball MT times VT final plus the final momentum of the golf ball mg times V G final and it's good to keep track of your unknowns right now I don't know the final velocities I'm given all these initial values of the velocity and the masses so those are those we can consider given up here the things I don't know are the final velocities so again just like before when we did this numerically I can't solve for either one of these because there's two of them so oftentimes what you do when you're solving problems symbolically since you're going to want to clean them up at some point you try to look for any ways you can simplify there aren't many ways we can simplify but I can bring this MT term over to the left and bring this mg term over to the right so I could write this as MT times VT I minus M T times VT final and similarly on the right hand side I'd get mg V G final and then I'm subtracting this term from both sides so I get minus mg Viji initial and you notice we can pull out a common factor you might be like why are we doing this well if I was doing this for the first time I wouldn't necessarily know either but it's often good practice to try to simplify as much as possible and in this case this is going to be crucial this is going to be an important step to trying to simplify this entire process right now I can see how that wouldn't be obvious but you got to just trust in me for a minute we're going to want to do this because it make our lives much better here in a second and on the right-hand side I'll write it as mg times V G final minus V G initial so just so you're keeping track the variables I don't know are this V T final and this V G final so I'm still stuck on this right-hand side it looks a little nicer because I've got terms grouped up but I'm still if this collision is elastic going to have to use conservation of kinetic energy so we'll do this over here so if I take the total initial kinetic energy and I set that equal to the total final kinetic energy I'll have one-half mass of the tennis ball the TI the initial velocity of the tennis ball squared so it's really just the initial speed of the tennis ball squared plus the initial kinetic energy of the golf ball which would be one-half M G V G I squared this is going to have to equal the final kinetic energy of all the objects so I'll have a one-half mass of the tennis ball final velocity of the tennis ball squared plus one-half mass of the golf ball final velocity of the golf ball squared and again we're doing this to clean this up we want to get a nice expression at the end so I'm going to cancel some terms look at one half sand everything so I can cancel one half from every term here by just dividing both sides by a half or I can imagine multiplying both sides by two that would get rid of all these 1 halves and then I'm going to do the same trick I played over here I'm going to get all my MT terms on one side and all my mg terms on another side again it might not be obvious why we're going to do this but I'm telling you something magical is about to happen so you got to take my word for it if I write em T times V T I squared and then I'm going to subtract this term from both sides to get the mts together so I'll have a minus M T V T final squared and that's going to equal because what I'm going to do is subtract this term from both sides so I get the M G's together I've got this term over here already M Viji final squared - and then this term that I'm subtracting from both sides M G V G initial squared and I play the same game I played over here I pull out a common factor it can pull out a common factor of M T I get em T times V T I squared minus V T final squared and that's going to equal M G times the quantity V G final squared minus V G initial squared and now things are getting interesting if you look at this left equation and this right equation they're looking a lot more like each other which is great because what I want to do is plug this right-hand equation into the left hand equation in a clever way that causes things to cancel that's why I'm doing it this way I mean we could have done brute force just like we did numerically solve for one of these velocities plug it straight into one of these velocities get a huge mess and try to like combine terms but this way we're doing it right here is going to be much cleaner so what do we do at this point I want to make this left-hand equation look more like this right-hand equation so is there any way I can change this difference of squares into a difference of just velocities and there is if you remember I can write this squared term I can write this as M T times the quantity V T I minus V T final multiplied by V T I plus V T final because when I multiply these together I mean get V T I squared minus V T final squared and then the cross terms are going to cancel if you don't believe me try it on your own check to make sure this multiplies out to get that and it will so I'm going to replace this term with this and that's going to equal I'm going to do the same thing on the right-hand side I'm going to write this as mg times the quantity V G final minus V G initial and then that multiplied by the quantity V G final plus V G initial and again this will multiply to give me that if you don't believe me try it out on your own and now we're in business check this out this is where the magic is going to happen I've got MT V TI minus VT final here and I've got that exact same expression over here so what I'm going to do is take this entire expression mg times the quantity of V G final minus V G and you'll I'm going to plug it in for that and you might be like how come what why can we do that well it's because this term here MT times this difference is the exact same as this term here MT times this difference and I know that this term equals that term they're the same thing I can replace this anywhere I see this I can replace it with that because they're equal so that's allowed I don't affect the Equality if I just plug this term in for that term because they're equal so that's what I'm going to do I'm going to take this term for mg I'm going to plug this all the way in over to this hand side and what am i what am I going to get I'm going to get mg times VG final minus VG initial so that's what this whole term equals but it's still multiplied by that so I got to bring this down and still multiply by this one which is V T initial plus VT final and that's going to equal this right hand side just stays the same I didn't do anything to that and now do you see it do you see how wonderful this is I can divide both sides by mg that cancels out and that's kind of weird there's going to be no mass left in this expression that we find so the relationship we're about to get doesn't depend on the masses of the objects colliding which is a little weird and cool but even better look at this term VG final minus VG initial that's right over here VG final minus VG initial so I can divide both sides by that and that cancels out and we're going to get one of the simplest expressions you could imagine let me make some room for it let me clear this up we're going to get that VT initial the initial velocity of the tennis ball plus VT final the final velocity of the tennis ball has to equal VG initial I'm going to switch the order here because they're adding and you can switch the order of things that you're adding just so it looks like the left hand side VG initial the initial velocity of the golf ball + VG final the final velocity of the golf ball look at how beautiful this is it says that in an elastic collision if you take the initial and final velocity of one of the objects that has to equal the initial plus final velocity of the other object regardless of what the masses of the objects colliding are and I would have never seen this unless we would have solved this symbolically to see that stuff cancels would not be obvious I could have solved a million of these elastic problems and probably never would have guessed that this was the case and the big reason why this is useful is because now we can use this simple expression as opposed to using conservation of kinetic energy conservation and kinetic energy was the thing that was giving us all the problems because it had the square of the speeds and when we plugged an expression in and squared it we got this nasty algebraic expression that we had to deal with but now with this simple expression between velocities we can simply solve for one of these unknown velocities in this equation and plug it into the conservation of momentum equation there will be no squaring of an expression you're still going to have to plug one equation into the other but the process will be much cleaner much simpler and much less prone to algebra errors and in the next video I'll show you an example of how to use this process to quickly find the final velocity of either object in an elastic collision so recapping we use the symbolic expression for conservation of momentum plug that into the conservation of energy formula and ended up with a beautiful simple result that we're going to be able to use to solve elastic collision problems in a way that avoids having to use conservation of energy every single time