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.
