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Current time:0:00Total duration:10:20

How to use the shortcut for solving elastic collisions

Video transcript

all right here's pretty much the fastest way you can solve one of these elastic collision problems when you don't know two of the velocities in this case we don't know the final velocities we know the initial velocity of the tennis ball and its mass we know the initial velocity of the golf ball and its mass but we don't know the final velocities of either ball and the trick to make these calculations go faster for an elastic collision is to use this equation which says that the initial velocity of one of the objects before the collision plus the final velocity of that same object after the collision should equal if it's an elastic collision it'll equal the initial velocity of the second object before the collision plus the final velocity of that second object after the collision if you want to see where this comes from we derived it in the previous video but now that we know it we can just use it for any elastic collision so this expression right here only holds for elastic collisions but you should love this formula right here because what this does is it allows you to avoid having to use conservation of energy and it avoids having to square terms and create horrible messes of algebra this formulas going to be much cleaner much nicer let's see how to use it let's just say object one is the tennis ball and object two here is the golf ball so in that case the initial velocity of the tennis ball that'd be 40 and it'd be positive 40 I'm going to write positive just so I know that's to the right and it should be positive plus the final velocity of the tennis ball I'll write that as VT so instead of writing 1 I'm going to get confused which one was number 1 so I'll write a VT that way I get to know which object's velocity I'm talking about so VT final that's what VT final is going to mean final velocity of the tennis ball and that should equal the initial velocity of the second object our second object is the golf ball the initial velocity of the second object of our golf ball is not 50 it's negative 50 you've got to be careful these are velocities in this formula so if you've got a velocity that's directed in the negative direction you better make it negative and if you solve in here you might get a negative so these are vector values up here you got to plug them in with the proper sign so this initial velocity of the golf ball would be negative 50 meters per second because we're going to assume leftward negative and rightward is positive plus the final velocity of the second object second object is our golf ball I'll call this VG instead of V to VG will be V of the golf ball and an F for final ie after the collision so we can solve this for VT final I can subtract forty meters per second from both sides and I'd get that the final velocity of the tennis ball is going to equal I'll have this VG final just sitting over here final velocity of the golf ball after the collision and then negative fifty minus of forty is going to be negative ninety meters per second so this formula alone was not enough because I've still got two unknowns I can't solve for either I've got to use another equation and the other equation we're going to use is conservation of momentum because during this collision the momentum should be conserved assuming the collision happened so fast any net external impulse is negligible so we can say that the total initial momentum is equal to the total final momentum we basically do this for every single collision because we make that assumption that the net external impulse during this collision is going to be small that means the momentum should be conserved so the formula for momentum is mass times velocity so the momentum of this tennis ball initially is the mass of the tennis ball point oh five eight times the initial velocity would be positive 40 positive because it's directed to the right and I'm going to consider rightward is positive plus the initial momentum of the golf ball would be 0.04 five times the initial velocity of the golf ball and that'd be negative 50 again you have to be careful with the negative signs momentum is also a vector so if these velocities are ever negative you've got to plug them in with their negative sign and that initial momentum should equal the final momentum so the final momentum of the tennis ball is going to be zero point zero five eight times our final velocity of the tennis ball and I'm going to use the same nomenclature I'm gonna use the same symbol over here that I used over here this VT final final velocity of the tennis ball is the same as this VT final final velocity of the tennis ball plus 0.04 five times same thing final velocity of the golf ball I'll use the same symbol which is going to be V G final I could multiply out this entire left-hand side and what I get is zero point zero seven kilogram meters per second and that schools on the right-hand side this entire expression right here I'll just copy that so we've got two unknowns in this equation as well so we can't solve this directly for either of the final velocities but we do have two equations and two unknowns now and whenever you have that situation you can solve one of the equations for one of the variables and plug that expression into the other equation in other words I know that the final velocity of the tennis ball is equal to the final velocity of the golf ball minus ninety so I could take this entire term right here since it's equal to VT final and just plug that in for VT final and what that would do for me is give me one expression all in terms of the final velocity of the golf ball so let's do that we've still got zero point zero seven kilogram meters per second on the left and that's going to equal zero point zero five eight that's still here kilograms times VT final I'm plugging in this entire expression for VT final so that gets multiplied by VG final the final velocity of the golf ball - ninety meters per second that was the term I plugged in for VT final and it got multiplied by this mass here so I can't forget about that mass and then I still have to add this final momentum of the golf ball we know for five kilograms times the final velocity of the golf ball so at this point you might be feeling ripped off you might be like easy way to do this this isn't easy this is hard I've got to plug one equation into another and then solve well this easy approach does not avoid having to plug one equation into the other that's true but the reason that it's easy is because the equations that we're plugging into each other are a whole lot simpler than the kinetic energy formula that you would have to use if you didn't know this expression here because we have this one we do not have to plug conservation of momentum into conservation of energy that would square the term we put in now we get nasty the algebra would be a lot worse these formulas that we're dealing with in this process only have velocity none of these velocities are squared so the algebra doesn't get nearly as bad and we're actually almost done over here let me show you how close we are to finishing this thing I just need to multiply through this term right here so what I'm going to get this left-hand side stays the same and then when I multiply through 0.058 I'm going to get zero point zero five eight times the final velocity of the golf ball and then negative ninety times point zero five eight is negative five point two two and that would be units of kilogram meters per second and I still just have this term right here so I'll copy that one and now here's a key step the whole reason we plugged one formula into the other is so that we had the same unknown in that formula only one unknown there's only one unknown variable in here which is the final velocity of the golf ball it's located in two spots but at least it's the same variable what that allows us to do is to combine these terms if on this right-hand side I have this much of the final velocity of the golf ball and that much of the final velocity of the golf ball when I add them up I can just add these two factors out front in other words I'll have zero point zero seven equals I can rewrite this as zero point zero five eight kilograms plus zero point zero four five kilograms times the final velocity of the golf ball and then I still have this minus five point two two so if that looked like mathematical witchcraft all I did is I combined the terms that had VG final because if you have a times VG final plus b times VG final that's the same as a plus B times VG final when you multiply this through you'll just get both of these terms back again so we keep going don't divide by this first sometimes people try to divide by this whole term right now you don't want to do that you got to go in the right order I need to add this 5.22 to both sides to get rid of it first so when I do that when I add 5.22 to both sides it will cancel this term and on the left hand side I'll get 5.29 kilogram meters per second and that's going to equal if I add these two terms together if I just add point oh five eight and point oh four five I get zero point one oh three kilograms times the final velocity of the golf ball and let me move these initial velocities down so we can see at this point if I divide both sides by 0.103 I'll get 51 0.36 meters per second on the left hand side and that equals the final velocity of the golf ball so we did it we found one of the final velocities of these objects we found the final velocity the golf ball but what about the final velocity of the tennis ball how do we figure out what that is because that's also an unknown in this problem well that one's really easy now that we know the final velocity of the golf ball I can just take that value plug it right back into here this expression that we plugged into conservation of momentum and I can figure out what the velocity of the tennis ball is so in other words the velocity of the tennis ball I know finally should equal the final velocity of the golf ball which was fifty one point three six meters per second and then minus ninety meters per second and I'll get that the final velocity of the tennis ball was negative thirty eight point six four meters per second and it came out to be negative that means this tennis ball got deflected backwards it was heading leftward thirty-eight point six four meters per second after the collision so recapping we use this nice formula to get one equation that involved the velocities that we didn't know for an elastic collision which you can only use for an elastic collision if you want to see where this equation comes from we derived it in the last video we solved for one of the velocities and plugged it into conservation of momentum we combined the like unknowns and solved algebraically to get one of the final unknown velocities then we plug that back into the first equation to get the other unknown velocity and this is easier than the alternative because the alternative involves kinetic energies which means when you take one of these expressions you'd have to square them and the algebra would be significantly more difficult