11. Momentum & Impulse
Intro To Elastic Collisions
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Hey guys. So in some problems you'll see that objects collide but they undergo what's called an elastic collision. Remember that? This is one of the two sort of broad types of collisions that you could have. What I'm not showing this video is that like all collisions, we're going to solve these kinds of problems by using conservation of momentum. But for elastic collisions we're also gonna need a special extra equation to solve them. Let's go ahead and check this out. Remember that? The whole idea is that momentum is always conserved regardless of the type of collision, whether it's elastic or inelastic. But the elastic collisions have a special sort of characteristic, which is that they also conserve kinetic energy. So the kinetic energy is also conserved. So K initial is K final for the system. So we're gonna use this characteristic right here to actually help us solve our problems. Let's go ahead and work out this example together, we'll come back to this in just a second. So the idea behind this problem is that you have these two blocks that are smashing into each other. You can kind of think of this as like two billiard balls that are crashing into each other. So we have this one that's going to the right, this one's going to the left, and we want to do is we want to calculate the final velocities of both the blocks after the collision. So we're gonna go ahead and stick to our steps. We're gonna need diagrams for before and after. So let's go ahead and draw that right. So afterwards, after these things collide, we're gonna have this three kg box, this five kg box, and we want to figure out their final velocities. So if I call this object one and two, what we're really looking for is V one final and then V two final. So basically these are our target variables like this and that brings us to the second step. We're gonna have to write our conservation of momentum equation and we'll come back to this in just a second here. So our conservation of momentum is M one V one initial plus M two, V two initial equals M one V one final plus M two, V two final. Right? So we have our masses and some of the speeds, let's go ahead and start plugging in our numbers. So we have five plus three equals five plus three. And now we have the speeds right? So this five kg block is initially going to the right at two. So I'm gonna plug into this one is going to the left at four, so I'm gonna plug in negative four. And then these two final velocities are actually our target variables. Remember this is what we're looking for here. So I can simplify the left side because I have all the numbers, This is just 10 -12. So you get negative too, so negative two equals five. V one final plus three V two final. So we still have two unknowns on the right side of this problem here. Right. And you can't assume that the V one final, V two final are the same because that doesn't necessarily happen in elastic collisions. So what happens is whenever we end up with an equation with two unknowns, we're gonna need another equation to solve it. And that's what's special about elastic collisions. For elastic collisions only, we often must use an extra equation, which I like to call the elastic collision equation. It goes like this V one initial plus V one final equals V two initial plus V two final. This really is sort of like the mathematical consequence of this conceptual point right here, which is that the kinetic energy is the same. Your textbooks will derive this, but you don't really need to know the details. So all you need to do is just memorize this equation. And there's a couple of really important things about this equation, which I like to talk about here. So the first has to do with sort of like the order of the variables and how it looks similar to the conservation of momentum equation. Here's what I mean. So you're conservation of momentum has the ems and it also has 1212 It goes initial, initial, final and final. The elastic collision equation has no masses. And if you look at the letters and the numbers, it goes 11 and then initial, final, initial final. So the order of the variables is different. What I want you to remember here is that conservation of momentum goes and the elastic collision equation goes 11 2 2. That's a really easy way to remember it. Right? So the second thing I want to point out here about this equation is that this equation actually has the same unknowns as the one that equation that we got stuck with. So what happens is this equation here also has V one final and V two final here. So because we have two equations that have the same unknowns, we're going to end up with what's called a system of equations. Remember we have system of equations, there's a couple of different ways we can solve for them. And the easiest way to solve this is by using equation addition. So, here's what I'm gonna do. I'm gonna write up my mic elastic my elastic collision equation over here. So I'm gonna have V one initial plus V two initial. Sorry, it's the one final, so V one final equals V. Two initial plus V. Two final. So right on my elastic collision equation, What I want to do now is I want to solve this system of equations by using equation edition. Remember that equation edition, you're gonna line up the equations top to bottom so that you can cancel out one of the variables and eliminate one of them. If you eliminate one of the unknowns, you're gonna only gonna be left with one unknown variables. What I want to do is that basically want to put another equation down here so that I can add them together and eliminate one of the variables. So what I wanna do is I want to start off with this elastic collision equation and I eventually want to get it down here in a form where I can add them and then cancel out one of the variables. So let's go ahead and take a look here. We know these initial velocities. So I know this view in initial is to so two plus V one final equals and this is just the negative four, right? This V two initial. So this is a Plus V two final. So if you take a look here, I have a number and then V one final and then V two final. So I want to make this equation look the same way. So I'm going to move the numbers to the left and I'm going to move the V one final to the rights which end up with is six equals negative V one final plus V two final. So notice how I'm almost done because I have a number of the one final and a V two final. The problem is if I just plug it in the way it's written, I'm not gonna be able to cancel one of the variables because I have five and a three attached to these variables and I have no numbers here. So I'm gonna have to do is I'm going to have to multiply this equation by some numbers that I can get rid of one of the variables. Hopefully you guys realize that I eventually want to cancel this negative V one final and this positive VV one final because they're opposite signs. When you add them together, one of them is gonna cancel. So what I want to do is I want to make the number over here, be the same as the number over here. So we're gonna have to multiply this number by five. And when you end up doing is when you multiply this whole entire equation by five on the left side, you're gonna get 36 times five equals this becomes negative five V one final, right? Plus five V two final. So hopefully you guys realize that now when we add these two equations straight down from top to bottom, The V one finals are going to cancel out. And so all you're gonna be left with is the V two final, which you end up with is 28 on the left side and then eight V two final. And so when you go ahead and saw for V two final, you're gonna get 3.5 m per second and that's one of your target variables. So that is the final velocity of this three kg block. It's going to the right At 3.5 m/s. So now we figured out one of our target variables, how do we figure out the other one? This V one final. Well, remember that whenever you saw the system of equations and find one variable, you're gonna plug that first target variable into your other equations to solve for the other targets. You're basically going to plug this number into either one of these equations. It doesn't matter which one you use to get the one final. So I'm gonna start off with this one over here because it's the easier one, the numbers a little bit nicer. So I'm gonna bring this down here and I'm gonna have a negative two equals five V one final plus three. And I actually know what this V two final is. I just saw for that, it's 3.5. So now what happens is when you move all the numbers around, you're gonna get negative 12.5 equals five V one final. And what you'll get when you solve is V one final is equal to negative 2. m per second. So, notice how you got the other target variable by plugging into either one of your equations. So now this five kg block is actually moving to the left final and its velocity is negative 25 meters per second. All right. So that's how you solve. You just write your conservation of momentum, your elastic collision equations. You're gonna solve a system of equations for those two unknowns. All right. So the last thing I wanna point out here is that you might actually not see this elastic collision equation written like this in your textbooks. Sometimes the orders will be different. You might see some minus signs or something like that, but I highly recommend that you or that you memorize it this way, because I think it's the easiest way to learn. All right, that's it for this one, guys, let's move on.
Head-On Elastic Collision
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Hey everyone. So hopefully got a chance to work this problem out on your own. So you've got these two blocks of equal mass they're gonna undergo and head on elastic collision and we want to calculate the magnitude and direction of the final velocities after colliding. So let's go ahead and work this out one step by step. So the first thing we do is just draw a quick little sketch of what's going on. So you've got these two blocks that are heading towards each other. So I'm gonna call this one block A. And this one block B. We've got that V. A initial is equal to five. And then we've got VB initial like this is equal to negative three, right? It's to the left, so it picks up a negative sign and ultimately they're going to collide. And then afterwards we want to figure out basically where they're going and how fast. So after the collision block A and B, they're gonna have sort of unknown speeds, right? They could be both going to the right or they could be going off in different directions or to the left. So we can't draw any arrows, but basically we want to calculate what are the final velocities of these two blocks. So let's move on to the next step, we're gonna write our both of our equations for conservation of momentum and elastic collisions. Remember ultimately we want to solve a system of equations, we're gonna have to write both of these. So for the conservation of momentum, this is going to look a little bit different than this because instead of M one V one we're really gonna have M A V A. Right? That's the object. M A V A initial plus MB Vb initial equals M A V A. Final plus M A M B V B final. Right? And then let's see. So I'm just gonna and then over here we've got is we've got our elastic collision equations. Remember this is the only one that we can use. We can only use this equation for elastic collisions. And it's basically that we have V one initial plus. Actually I'm gonna call this V A initial plus V B initial equals or sorry, via final equals Vb initial plus vb final. Remember they look different +12121122 and so on and so forth. Okay, so we're basically just go ahead and start solving the system of equations by plugging in some numbers. So what happens is in the first problem here, usually we would just start plugging in the masses of each of these objects, but we actually don't know what they are but that's fine because the mass of A is actually equal to the mass of B. Because they're equal mass. Alright, so they're equal mass. So that means that Emma is equal to M B. What that really means here is we can actually cancel out the m term from the whole entire equation. Remember if you have the same number that goes through all of your terms, you can just cancel it out completely. In other words the mass actually really doesn't matter in this problem or at least in the equation. So so then let's go ahead and start plugging in some initial values. So via initial is going to be the five. VB initial is going to be the negative three. This equals V. A. Final plus VB final. And when you simplify this, what you're gonna get here is you're gonna get two equals V. A. Final plus VB final. Alright, this is the first equation that we're going to need to solve our system of equations because remember this is the one that has two unknowns. So let's look at the other equation. The elastic collision equations. Remember if we start plugging in our values via initial is five V A final is unknown. Vb initial is negative three and then VB final is unknown. Okay, so we also have these same two unknowns in this problem. And so we want to do is again we want to sort of add something to this first equation so that one of the terms will cancel out and then basically you're left with one unknown. Alright, so we want to sort of stick another equation down here um so that we can cancel out one of the terms. Okay, so what happens is when you bring the negative three over to the other side, it becomes a positive and you end up with eight and then when you move the V A final to the other side, it picks up a negative sign. So in other words you get eight equals negative V A final plus VB final. Alright, now again this is where we would have to either multiply this equation to get one of these terms to cancel out, but we actually don't have to do that here because in this equation we have a V A final and we also have a negative V A final in this equation. So we don't actually have to multiply this by anything just to make the numbers line up so we can actually just go ahead and stick this equation right down in this box. So we're gonna add this to let's see this is eight equals negative V A final plus VtB final. Notice how now when you stack these two things on top of each other and then you add them down. The V A finals will just cancel out. And what you'll end up with here is you'll end up with 10 on one side and you'll end up with two vB final on the other. Alright, so now that we've sort of eliminated one of the equations, we're just gonna go ahead and solve. So this final velocity here, this v be final for B is going to be five. So now let's move on to the last step here, which is this is one of our target variables. This Vb final here is actually going to be going off to the right like this. So this VB final equals five and we got a positive number so it points to the right. The last thing we have to do is we have to plug the first target variable into any of the other equations to then solve for the other missing variable. So in other words we can stick this VAVB final into either one of these two equations to solve for the other one. It really just your preference, they're both pretty much the same. I'm just gonna go ahead and go with the first one here. So then basically if we write rewrite equation number one, you're gonna get is the two is equal to V A final plus VB final but we actually know that that's five already. So in other words this is just gonna be five like this. Okay, so um we just go ahead and solve and we're just going to get that V A final is equal to negative three m per second and that is your other final answer. So in other words, V A final is going to be moving to the left at negative three. Alright, so now I want you to notice that something interesting happened in this problem. So we actually have these two blocks of equal mass they collided and what happens is the via initial for object A was five and then the VB initial for object B was negative three. But afterwards their velocities basically switched. So now V. A final is negative three, whereas VB final is five. So in other words, the V A. For object A became the VB the final velocity for object B and vice versa for objects for the for the second block. So in other words, when you have these two objects of equal mass and they undergo an elastic collisions. One pro tip that you can use is they actually trade or exchange velocities. So they trade or exchange, basically they just swap the initial and final velocities between the both objects. Alright guys, so that's it for this one.
Special Equations in Elastic Collisions
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Hey everyone. So now that we've covered the basics of elastic collisions, there's a special type of problem that you'll see, which has a very common setup, it's where one moving object like this sliding block over here is gonna hit and collide with a stationary object like this one over here. And the basic idea of these problems is instead of having to use a system of equations to solve their final velocities, We're actually gonna be able to use these special equations that I'm gonna give you in just a second down here to solve them. All right. So for these problems only we can use these kinds of special equations. So let's go ahead and get started and we'll just jump right into a problem. The basic idea of these problems is that this stationary object here isn't moving, this M2 is never moving. And because of that, the initial velocity of object to is equal to zero, but that helps us do is simplifies our equations. So what your textbooks are gonna do is we're gonna do a derivation where they're gonna take the momentum conservation and elastic collision equations and they're gonna cancel out this term because it's zero. And then basically, through some substitution and some algebra, you're gonna get to these two expressions for the final velocities of both of the objects and I'm just gonna give them to you. The first one is you're gonna do em one minus M two over M one plus M. Two times V one initial. And the second one is going to be two M one divided by M one plus M two over V times V one initial. So what you'll notice here is that these two equations actually look very similar. They both have the total mass in the denominator and they're both multiplied only just by the initial velocity of the first object. And then the numerator is are a little bit different. Right? But that's basically it. You just use these two equations to find the final velocities of both of the objects. So let's just get right into our problem and see how we use these equations. So the basic setup of this problem is that we have a round boulder with a massive 40 kg and a golf ball with a mass of 0.1. And we're gonna calculate their final velocities of both of these objects basically after they collide for these three cases where we're gonna have the two masses that are the same and then we're gonna have the lighter one hitting the heavy one and then vice versa. So let's just get right to it. So in part a we have the boulder that hits another boulder. So in other words, we want to calculate V one final. And we're just gonna write this out a good way to remember this is that you subtract the masses and then you add the masses and then you multiply by the first, the initial velocity of the first object. So in other words, we're gonna take 40 minus 40 divided by 40 plus 40. Right? Because they're the same and they're gonna multiply this by the initial velocity of the boulder, which is five m per second. And in fact, it's always going to be five throughout the entire problem. So what happens in this in this equation is that you actually end up canceling out 40 minus 40 because it's zero and that it doesn't matter what all this other stuff is, because zero in the denominator is going to make everything equal to zero. So in other words, the boulder basically stops what happens to the second boulder? What's the final velocity? Well, our equation again is to M one over M one plus M two times view on initial. Some of the words, it's two times 40, divided by 40 plus 40, and then times five. So in other words, what happens here is you're going to get 80/80 and then times five, so really 80/80 is just one. And so this V two final here is equal to five m per second. So, those are are basically two of answers, we have that this boulder basically just stops, right? The first one, but then the second one is going to go off at five m per second. Now, this should make some sense because elsewhere in our elastic collision videos, we said that if two objects of equal mass collide, they basically just trade velocities the velocity of the first one becomes the final velocity of the second object and that's exactly what happened here. Let's take a look at part B and part B. Now we have that. The golf ball which is M one is going to hit the boulder which is the 40. Alright, so V one final is going to equal and I'm just gonna start plugging in the numbers here. So M one is going to be your 0.1 and it's really important that you plug them in in the correct order. The one that's moving is always going to be M one, The one that stationary is always going to be M2. So when you plug this into your equations here Keep track of that. 0.1 -40 divided by And this is going to be 0.1 plus 40. And then times five When you work this out, what you're gonna get, you're gonna get a negative number, you're going to get negative 4.98 m/s. Alright, so that's the first one. The 2nd 1, V two final Is going to be, it's going to be two times 0.1 divided by 0.1 plus 40. Times five. So in other words, what you're going to get here is 0.02 m/s. So let's take a look at what happened. The golf ball after hitting the boulder basically just goes ricochets backwards at almost the same speed that it came in, but just negative it can it goes out with negative 44. m/s. The boulder on the other hand picks up a tiny little bit of speed and it goes off to the left to the right at 0.02. This should make some sense because when the golf ball hits the boulder, it transfers a very little amount of momentum. It's going fast, but it has a very little mass, whereas the boulder has a lot of mass, so it only picks up a little bit of speed. But the golf ball basically just ricochets backwards at almost the same speed at which it came in with. All right, So, let's take a look now, our final answer, our final part, which is where the boulder now hits the golf ball here. The situation is reversed because now what happens is that your golf ball is M2 and your M1 is going to be the boulders. That's the most important thing. So, your V one final is going to look like M one minus M two. So now it's going to be 40 minus 0.1, notice how it's reversed from this from part B over Plus 0.1, Times five. Now, when you work this out, what you're gonna get is that this is 4.98 m per second. V two final is going to be two times 40 divided by 40.0 plus point point 01 and then times five when you work this out, you're gonna get is 9.98 m per second. So now let's look at what happened here here. What happens is that the boulder, when it hits the golf ball, it loses very little momentum, so it's still traveling to the right at almost the same speed at which it hits. But now the golf ball has picked up a ton of speed from the boulder. Once the boulder smacks into the golf ball, it goes off at a much higher speed 9.98 m per second. All right, so, I just want to sort of summarize these cases. Uh He started limiting cases where the masses are equal and much less and much greater than. So, this is basically what happens whenever you have the masses that are equal, the final velocity, the first one is going to be zero and the final velocity of the second object is basically gonna be whatever the initial velocity of the first one was. That's exactly what we saw here with the boulder to boulder case, you might remember if you ever played a ball game of billiards when you hit the cue ball into another ball, the cue ball basically stops and then the other one just goes off with the same speed at which you shot the white ball with the cue ball. Alright, so that should make some sense if you've ever played the game of pool. So for the second, for the second case where one object is much much less massive than the other one. In other words you have a very massive target. What happens is that the the final of the first one is going to be the negative the final the initial of the first object, right? It basically just goes back with as much speed but just in the opposite direction. Whereas the second object, if it's much much much more massive, what you'll see is that this is basically just equal to zero. It picks up a very very little amount of speed now for the final, basically what happens is that the V one final is going to be pretty much whatever it hit with, right? Because the massive objects, massive projectile isn't going to lose a whole lot of speed, whereas the final of the second is actually gonna be two times V one initial. That's exactly what we saw with the golf ball. It picks up basically a ton of speed, that's almost double of the five m per second that it came in with. So the very last point I want to make is that after the collision the second mass is always going to move forward, right? Whatever is being hit is always going to move forward just a little bit, but this m one might move forward or backward depending on its mass. What we saw here is that one could either stop. It could also go backwards or forwards, and it really all depends on the mass. Alright, so that's it for this one, let me know if you have any questions.