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Anderson Video - Momentum in 1D Car Collision

Professor Anderson
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 >> Alright, let's talk a little bit more about this very simple equation. Momentum initially equals momentum finally, and let's say that we have two particles that are going to collide, okay they're going to interact somehow; maybe they collide, maybe they explode, maybe they just have some forces that are acting on each other. How do I describe what we just said here in terms of these two particles? Well Pi is going to consist of two particles and two components, x and y. So on the left side we really have four quantities and on the right side we're going to have four quantities. And so this is where it gets a little cumbersome with the subscripts but the way I do it is this, P1 initial in the x direction plus P2 initial in the x direction is going to be equal to P1 final in the x direction plus P2 final in the x direction. That's the first component of this equation with our two particles, one two. The second one is therefore P1-- sorry. P1 initial in the y direction plus P2 initial in the y direction is equal to P1 final in the y direction plus P2 final in the y direction. Okay it looks like a lot to handle but it's really not so bad; x components, y components, initial on the left, final on the right, two particles, one and two. Alright. So let's take a look at a simple problem of a car collision and let's do the following; M1 and M2 collide head on and they come to a stop. Okay we're going to say this is our initial condition. And let's draw a picture of this thing and when you draw a momentum picture you always want to have a before picture and an after picture, so here's our before picture, and then the after picture, they've come to a stop, we know if they're going and then the after picture, they've come to a stop, we know if they're going to have a head on collision there's going to be some deformation of the metal, so these things will be bent up and everything will come to a stop; V final equal to zero. And let's say that this is mass M1, this is mass M2, this is V1 initial, this is V2 initial; these are the speeds of the two cars. Let's attack this from the idea of conservation of momentum. Now when you look at this problem is this a one d problem or a two d problem? It's a one d problem, right, everything's coming along in a line, two cars coming straight at each other, they collide and that's it. So I can think about my system as those two cars. Have I really created an isolated system? Yes and no, right. It looks pretty good but we know that the cars are stuck to the ground because of gravity so there's a force crossing our dash line, and we know there's a normal force pushing back up on the cars. So the reason that we can make this our isolated system is because gravity down is exactly canceled out by the normal force up. Those two cancel each other out and therefore they're not going to come into play in this problem. There's no movement in the y direction, it's all one dimensional in the x direction. So we can simplify our conservation momentum equation to just one dimension, P initial in the x with P final in the x. P initial in the x is consisting of two quantities. We have mass one times velocity one, which is positive, it's going to the right in our picture. And then we have mass two times velocity two and that one we put a negative sign on because it's going to the left in our picture. And all of that is going to be equal to the two cars stuck together now, which happens in a perfectly inelastic collision times V final. But our initial condition was V final has to be equal to zero, okay that's what we gave you in the problem. And so this is gone and that whole thing equals zero. And so now look what happened, we have a very nice simple relationship between the two. If the final momentum of the whole system is going to be zero that means the initial momentum of the whole system has to be zero and therefore the momentum of each car has to be equal and opposite, alright; they have to be the same magnitude M1 V1 initial M2 V2 initial and they have to be pointing in different directions. Okay and now with this relationship you can see what happens. Obviously if M1 is equal to M2 then the only way we're going to satisfy this is if the speeds are equal, and that should make sense to you, right; two cars collide and they're the same mass if they're going to come to a stop they have to be moving at the same speed. But let's say that M1 is very big; let's say that M1 is a truck. If M1 is a truck and the whole thing is going to come to a stop and M2 is a small sports car, which one of these has to be moving faster? Kevin. >> The car has to be moving faster. >> The sports car has to be moving faster, right. We can solve this equation very quickly for V2. V2i is going to be M1 over M2 times V1i, and if M1 is a truck and M2 is a sports car then this number right here is bigger than one and therefore V2i has to be bigger than V1i. The sports car has to be moving faster than the truck in order to get it to come to a stop, and the smaller the sports car is the faster it has to be going in order to get that truck to come to a stop. Okay. Alright, excellent. Anymore questions about head on collisions? Don't-- don't do this at home. Not a good experiment to try.