Momentum conservation - Solved example

- [Instructor] Let's solve two problems on collision. Here is the first one. A nine kg monkey jumps with a horizontal speed of ten meters per second onto a stationary one kg skateboard. With what speed do the monkey and the skateboard move together? Neglect the friction between the skateboard and the ground. So, let's look at what is given to us first. Maybe we'll draw a diagram for that. Then, we'll gather the data, we'll see what is asked, and then, maybe, we'll come up with a strategy to solve that. Okay? So, let's first go ahead and look at what's given to us. We're given that a monkey is jumping onto his skateboard, so here is our picture. So, this is our monkey. He's gonna jump on that skateboard, and then they'll start moving together. What data is given to us? Let's look at that now. We know the weight of that monkey. We know the mass of that monkey is nine kilograms. We also know the mass of that skateboard is one kilogram, so let's put down these masses. I'm not going to write those individually, that data, individually. To save space, I'm just writing it on this picture. And we're also given... What else? We are given the monkey jumps with a horizontal speed of ten meters per second. So, this monkey has a speed of ten meters per second, and the skateboard is stationary. So, the skateboard, initially, is at rest. Now, what we need to find out? What do we have to calculate? With what speed do the monkey and the skateboard move together? So, that means after jumping on that skateboard, we need to figure out what is the speed with which they move together, so their combined speed is what we need to calculate. So, how do we solve this problem? Where do we even start? Well, I guess, the most important clue that we can get over here is that this is a problem on collision, and whenever bodies are colliding, their total momentum before collision should always equal their momentum after the collision, and so that's where we can start. And, just to make that statement a little bit more clear before we start solving, what's collision? Well, in physics, whenever objects put a force on each other for a short time, we say they are colliding. For example, when the monkey jumps on this skateboard, it puts a force on that skateboard. It pushes that skateboard forward, which is what accelerates the board, and during the same time, the board pushes back on monkey, Newton's third law. But these forces only last for a short time. After which, they both start moving with a constant velocity, and therefore, that's a collision and momentum is conserved. And another important thing is that momentum is only conserved provided there are no external forces. If there are other objects besides these two, that start pushing on them, say, the ground starts pushing on them, then the momentum will not be conserved. It's for that reason it's mentioned in the problem, "Neglect friction," and if you're wondering, "Well, what about gravity? Isn't that an external force?" Well, yeah, but we don't have to worry about gravity, because it's been balanced. For example, if you consider the gravitational force acting on this monkey downwards, that's being balanced by an upward push given by the skateboard, and so, the forces cancel out, and we don't have to worry about them. So, if you neglect friction, there are no other external forces, and so, we can see the momentum is conserved. And of course, if you need more clarity on these things, then we have talked a lot about them in previous video, called, "Conservation of Momentum." So, you can always go back and refer to that. All right, now, I think we can start the problem, and we will start by saying that the total initial momentum, which we will write as Pi should equal the total final momentum, momentum after the collision. Now, we know how to calculate momentum, right? We just multiply mass and the velocity, so all we have to do is calculate the initial momentum of the monkey plus that of the skateboard, and equate it to the final momentum of the monkey and the skateboard, and then, see if we can calculate that velocity. So, you know what? Great idea: You have to give it a shot yourself. Pause the video and see if you can first try this yourself. Okay, so, let's go ahead and solve this. So, what's the initial momentum? Well, that's the initial momentum of the monkey which is the mass of that monkey. I'll use big-sized letters for monkey, and I'll use small-sized letters for the skateboard. I'll also use different colors for them. So, that's when we mass of that monkey into the initial velocity of that monkey plus mass of that skateboard into the initial velocity of that skateboard. This is the total initial momentum, and that should equal it's total final momentum. What's that? Again, that's gonna be mass of that monkey into the final velocity of that monkey. Notice the final velocity's the same for both of them, because they're moving together. Plus, the mass of that skateboard, the mass of that skateboard, into the final velocity of that skateboard, and before we substitute, since v is common, because they are going with the same velocity, we can pull that v out and write is as the total mass, M + m into v, and now we can plug in. We have all the data, and we can plug in. So, let's do that over here. The mass of the monkey is nine kilograms So, that's going to be nine kilograms times u, which is ten meters per second, plus the initial velocity of our skateboard is zero, because, initially, it was at rest, so this whole thing goes to zero, so that's our left-hand side. This is our total initial momentum. That should equal - let's look at the right-hand side - it is M + m. That is the mass of the monkey plus the mass of that skateboard, which is nine plus one which is ten. So, let's use a neutral color for that. So, ten kilograms, the total mass, times v, which we need to calculate. ... And look, we can now calculate what v is, doing some algebra. So, the ten divides out, and the kilogram cancels, and we are done. V will equal nine, and the units left out is meters per second, and that's our answer. So, once the monkey jumps on that skateboard, the board takes off with a speed of nine meters per second. All right, let's solve one more. Can you pause and try and solve this yourself? Do it the same way. First, figure out what is given. Maybe draw a diagram, collect all that data, and then see if we can somehow solve this problem. All right, let's see what's given. We have a two kilogram toy truck moving at four meters per second is about to collide with a one kilogram toy car moving at 17 meters per second in the opposite direction. Find their combined velocity after collision, if they stick to each other. So, we have two toys coming in the opposite direction, so here is the diagram. And so, we have a toy truck and a car moving towards each other, and then they collide and stick to each other, which means after collision, they will move together, and we need to figure out with what speed and direction will they be moving together? So, we know their masses. We are given the truck weighs two kilograms, and the car has a mass of one kilogram. We also know their initial velocities. The truck is coming in at four, and the car is coming in at 17 in the opposite direction, and we need to find what their combined velocity is after the collision. Okay. Where do we start? Again, because this is a collision problem, we can start the same way we did before. We can see the total momentum of the car and the truck before collision should equal the total momentum after the collision, and again, just like before, because both of them have the same final velocity, we have pulled that v out from that equation. Now, all we need to do is plug in these values and calculate what v is. So, if you couldn't do this before, can you pause this one more time, and see if you can do it from here, and, remember, these two are coming in the opposite direction, so please take care of that. Again, give it a try. All right, let's substitute. So, the mass of the truck is two kilogram into the initial velocity of that truck is four meters per second, plus this time this object is not at rest; it's moving. It's mass is one kilogram, and now comes the important part. Can we say it's velocity is 17 meters per second? No, we can't. The main reason is velocities are direction sensitive. So, if this velocity is to the right, and we are taking that as positive, this velocity is in the opposite direction. So, if this is positive direction, we should call this negative, and that's very important, otherwise, we get the wrong answer. So, this is negative. Okay, when things are in the opposite direction, remember, one velocity is positive. One is negative. And that should equal their total mass which is M + m, that is, just three kilograms into v... And now if we simplify this particular equation, which I'm pretty sure you can do. So, I leave that thing to you just to save time. We will get V equals (if we simplify that) minus three meters per second And, again, you can pause and just verify that you get that answer, and that's our answer. So, this means after collision, they start moving at three meters per second, but what does that minus sign mean? Well, it's telling us about the direction. You see, since we took this as positive, and we took this as negative, that means we chose the right side velocity as positive and the left side velocity as negative. That's what we did, and since we're getting a negative answer, this now means that this whole thing will start moving to the left. If we had gotten a positive answer, it would have meant the whole thing was going to the right, okay? So, the final answer not only tells us the speed, it also tells us in which direction the whole thing is going, and so whenever we have problems with collisions, we can always start with momentum conservation.