**Concept:** Adding Mass to a Moving System

Hey guys. So, in this video we're going to talk about a special type of conservation of momentum problem, which is, when we have an object that's moving and then you either add another object to it or you remove an object from it. So, think of it as a cart and as the cart is moving you accidentally remove stuff, what happened then? Well, you have an interaction of two masses the cart is one mass and whatever you add or remove is another mass therefore is the conservation momentum problem, let's check it out. Alright, so as I just mentioned, if you add or remove a mass 2 refirming moving objects the speed of the system, which is the object plus the new mass may change, sometimes it does, sometimes it doesn't, it's a bit tricky but I'll walk you through it, the key thing to remember with all really conservation of momentum problems is that, if one object gains or loses the momentum the other object must gain or lose the same amount of momentum. So, for example, when you shoot a bullet the bullet moves this way and let's say the bullet gains a momentum of 100 then it must mean that this gun will lose 100 of momentum or gain negative 100 momentum and so that's the two cancel out. So, whatever whenever something's getting faster something else should get slower to compensate for that. So, I got three examples here, for us to look through different possibilities. Alright, so the first one, I have a 70 kilograms sled that is moving horizontally at 10 meters per second. So, I'm going to call this m1 equals 70 and v1 initial equals 10 meters per second. Alright, on frictionless ice. So, it's sled on frictionless ice, when a 30 kilogram child lands vertically on it, right? So, child just randomly Falls here, right? So, the 10, that's like a fat baby or something. So, 30 kilogram child, let's call that m2, is going to land here, doesn't get hurt, okay? Now, a child is a part of the system, before the sled was moving by itself. Now, it's sled plus child and I wanna know what is the sled's speed after the child lands on it and before we calculate this I want you think about it, do you think the sled gets heavier? I'm sorry, do you think that sled gets slower or faster? And I hope you're thinking that the sled gets slower because we're adding mass to it and the heavier you are the slower you are and that actually does happen. So, let's calculate this real quick, but before I do that one more thing, realize these guys, the sled is moving the x-axis, this is 10 meters per second in the x-axis, the child falls vertically, which means the child's v2 has some sort of Y velocity. Now, it's falling the y-axis but with the child falling vertically, the child isn't moving sideways, the initial velocity of the child child's change. So, V2 initial in the x-axis is actually 0, okay? The child isn't moving sideways, the sled is, and I want to know what is the sled speed after the child lands on it. So, I want to know what is v1, which is the sled final, and the sled is moving this way. So, we're explicitly going to put here, that we're talking about the x-axis, okay? I want to know what happens after. So, I'm going to write m1 v1 initial plus m2 v2 initial equals m1 v1 final plus m2 v2 final but in this problem since the child was originally moving the Y and the sled is moving the x axis, there are sort of two axes going on here. So, I'm going to explicitly say that all of this has to do with the x axis, okay? I want to know is moving at 10 to the x. Now, what's the speed in the x axis as well, okay? So, let's put the mass of the masses are 70 and 30 so I'm going to do this 70, 30, 70 and 30 and now, all we have to do is plugging the numbers and solve for the answer, before the child lands. Remember, initial always moves before some events, this is sort of a collision, right? Before the child lands this sled is moving with 10 the child, initially has a velocity of 0, talked about it right here, child had some velocity on the Y but has no velocity on the x axis. So, this whole thing is gone. Now, that the child lands on the sled and moves with the sled they will move together therefore they have the same final velocity, okay? They have the same final velocity, which means is it completely inelastic collision. Alright, so let's combine these here, I have 700 on this side equal, this is 100. So, I'm going to put the 100 here and the final velocity will be 70 meters per second, okay? I'm sorry, 7 meters per second not 70, which should make sense, remember we talked about, we should expect that the sled gets slower because you're adding a mass to it that in fact, happens. Now, let me just mention one more thing. So, essentially what's happening here is the child has no horizontal velocity, no horizontal momentum to be more clear and more precise and then once you land here, the sled is doing the work of getting the child from 0 velocity in the x axis to having some velocity in the x axis, so the sled gives momentum in the x axis to the child, the child gains some momentum, which means the sled has to lose some momentum and therefore some velocity, okay? Child gain velocity so the sled lost velocity okay, hopefully that makes sense, let's hop on the next one, I'm going to do this back-to-back because these are all sort of related and I want to really, I want to talk about them as a whole. Alright, so. Now, we have a boy that runs parallel to a 3 kilogram skateboard. So, I'm going to do a top view here, it's easier to see it this way, okay? So, imagine you're sort of like looking down and you see this boy running. So, I'm going to do this, this is going to come out terrible but imagine this is the head and these are the little arms, this is awful, but the child is running this way with 10 meters per second, which is totally unrealistic because that's like Olympic, Olympic athletes run that fast but anyway. So, m1 equals 40 and the initial velocity of 1 is 10, next the child is a skate skateboard. So, I'm going to draw a sort of a big skateboard these are like the wheels or whatever and he has a mass of 3. So, m2 equals 3 kilograms, just put a 3, and it's also moving with the initial velocity v2 initial of 10, okay? So, these two things at the same speed they're running next to each other, okay? They're going next each other like this and then the boy will step side way into skateboard. So, they're both going this way, the boy is just going to do this boom, right? While running, the boy is going to jump sideways, slightly, and they both continue to move in the positive x-axis, there's no changing direction, okay? So, the question is what is the new speed of the system, okay? So, somewhat over here you have skate with the boy on top and I want to know what is the final velocity, okay? This is sort of a collision in that these two objects were separate and now they came together, okay? So, what we're going to do is we're going to write the momentum equation, m1 v1, m2 v2, m1 v1, m2 v2, okay? And the masses are 40 and 3. So, let me do this, 40 and 3, 40 and 3 and the initial velocity of both of them is 10 and 10 and we want to know the final velocity, I don't know the first final velocity, I don't know the second find velocity but if the boy is riding in top of the skateboard, I know that he must be the same and look what's going to happen here, this is 700, actually it's not 700, it's 430, sorry about that, this is 430 equals 43 V final, V final equals 430 over 43, which is 10, what happened here is that, well, nothing happened really, the velocities didn't change at all neither one of the velocities changed, they were already moving next to each other the boy moves to the right because the velocity is already matched no one had to get faster or slower, if the skateboard was moving faster than the boy skateboard would have to get the boy to move faster, which therefore would mean the skateboard would slow down, right? There are already the same velocity, no one has to give anyone momentum. So, if noone gives momentum no one loses momentum and the whole thing stays the same, okay? Again, that's because you sort of step sideways into it, the next one is very similar and I'm going to draw a similar setup here, I got m1 equals 30 and I've got m2 equals 60. Now this is a sled and they're both moving v1 initial 15 actually the child is on top of this sled, sorry about that. So, let me put the little guy here, that's the child and this is the sled, and they're both moving with an initial velocity v1 an issue of 15, frictionless ice so there's no friction, the child falls sideways from the sled. So, what happens is a little bit later here's a sled, the child just sort of rolls over here, this way, okay? And we want to know what is a child's speed, so this is after and this is before. Now, the sled will obviously continue to move this way, sled is number two, so v2 final, you want to figure out what's that velocity and, but the first thing is to figure out, what is the velocity of the child v1 final, and this is sort of a conceptual question because you can't really calculate this, you just have to remember that when you come off a moving vehicle you inherit, you borrow the speed of that vehicle, the velocity of that vehicle. So, if the sleds moving this way with fifteen, when you come off the sled sideways you're going to keep going that way with fifteen, okay? That's just called inertia so this is actually going to be 15 to the right.

Now, check it out, if the boy was originally moving with 15 and you step side way and you keep 15, the boy didn't get faster or slower therefore the sled cannot get faster or slower either, so the sled keeps its 15. Now this is entirely conceptual for the second one I could have calculated it and I would have shown you that you just keep the same number kind of like how here we calculate it and we just kept the same number, okay? But trust me, if you calculate you get the same number and in both of these problems you get the same situation, which is since nobody or no objects or, I should say no mass because the person on objects, since no mass gains or loses momentum or, I should say since the boy doesn't gain or lose momentum then the skateboard or sled doesn't lose or gain momentum okay, if the boy doesn't gain momentum the sled doesn't lose momentum, if the boy doesn't lose momentum then the sled can't gain momentum, okay? So, think about it that way, if the, some people would have said that this got faster because you now remove the mass from the system, that's temptation, that's why this question is tricky, this is wrong, the only way that sled could have gotten faster is if the boy got slower, when he came off as a result of going away from it, right? For example, if the boy was here and the boy jumped backwards, the boy would be going from, let's say a velocity 15, and then if you sort of jumped backwards maybe his velocity would go from 15 to like a 5 so the boy went from going really fast this way to not so fast, lost some velocity. So, the sled would then gain some velocity in that direction, right? If the boy jumps this way the sled would have gone that way, okay? So, a little tricky but conceptually you need to understand how this works, let me know if you guys have any questions.