Calculating momentum changes - Solved example

- Let's solve a couple of problems on momentum changes. Here is the first one. A .5 kilograms ball of clay moving at 20 meters per second hits a wall and stops, sticks to it in .1 seconds. Find the change in it's momentum and the force exerted by the wall on it. Okay, so what do we do? Well first we'll try to think about what's given to us, maybe we'll draw a diagram, then we'll write down all the data, think about what is asked, and then think about how to solve it. Okay, so, what's given? What's going on? So we have a ball of clay that's moving at some speed, it hits the wall and comes to a stop. So let's first draw the diagram for that. Here is our ball of clay, it hits the wall, comes to a stop. Okay, let's see what's, what are the data given to us. Well we know that that ball of clay weights .5 kilograms so we know it's mass, we know it's coming in at 20 meters per second so it's initial speed, before hitting the wall, we know that it's 20 meters per second. We know it hits the wall and sticks to it, it stops, so that means after hitting, it's no longer moving it's stuck to it, so we know it's at rest and we also know, which I have it written over here but we also know that it takes .1 second to come to a stop. Alright, so before we go ahead, let's quickly write down all this data in one place. Let's do that. So, we know the mass, that is .5 kilograms, we know the speed of that clay, before it hits the wall. We usually call that the initial velocity and we usually use the letter U for that, so the initial velocity U is 20 meters per second. After it hits the wall, it's final velocity is zero because it's no longer moving, so we also know that it's final velocity is zero. And and we know that it takes .1 second for it to stop, for it to go from 20 to zero. So, time taken is also something given to us, .1 seconds. Okay, now let's think about what is asked. What is asked? We need to find the change in it's momentum and we have to calculate the force exerted, we'll talk about that a little bit later but we have to calculate the change in momentum and so I think the first thing we need to think about is what is momentum? Well we're talked about this in a previous video called Intro to Momentum, and the basic idea is, we calculate momentum as the product of mass and velocity. So you take the mass of an object, multiply by it's velocity and that number is called, the momentum. And we need to calculate the change in momentum. Ohh we see when the clay goes and hits the wall it's velocity changes, right? It goes from 20 to zero, so it's momentum must also change and we need to calculate how much that momentum has changed. And how do you calculate that? Well, change is always calculated as final value minus the initial value. Right? So we're here, the change in momentum would be the final momentum, which I'll call as PF minus the initial momentum. That's what is asked over here. Okay. So, can you first try this yourself. Give it a shot. Go ahead, see what the final momentum would be, what the initial momentum is, subtract and see what answer you end up with. Because we know how to calculate momentum, all that it has given, so go ahead, give it a shot. Pause the video. Alright, let's see. The final momentum would be, the mass into the final velocity, minus, the initial momentum would be the mass into it's initial velocity. And now if you plug in, the final velocity is zero, so the final momentum would just be zero, minus the initial momentum, that will be M that's .5 kilograms, .5 kilograms times U which is 20 meters per second, let me just squeeze that in over here, excellent. 20 meters per second and if we calculate that, let's see what we get. We get a negative, .5 times 20 that's 10, half into 20 is 10 so that's 10 kilogram, meters per second, let me put this in the same color. So that's our answer and if your wondering what does this negative sign is telling us, well it will make a lot of sense when we talk about the next part of the problem. The next part of the problem is asking us to calculate the force exerted by the wall on the clay. And if you think about it, when the clay goes and hits the wall, it comes to a stop that means the wall must be pushing on that clay. That's why it's coming to a stop, isn't it? So there is a force that the wall is putting on that clay and that is what we need to calculate, how much that force is. Now how do we do that? Well, whenever we are given the details of the motion of that object and we are asked to calculate the force acting on that object, I always like to go back to Newton's Second Law, which is, one of the most famous equations of physics, force equals mass times acceleration. But in this particular problem we can also use another equation for force. F equals change in momentum divide by time. We have seen in a previous video that these two equations are identical, you can derive one from the other and if you need more clarity on this particular equation, we've talked a lot about that in a previous video called Newton's Second Law and Momentum. So you can go back and check that. Now, anyways, Why, why should we use this one? Because over here the acceleration is not given to us directly. So to use the fast equation I have to calculate the acceleration and then plug it in, which we can do of course, we can do it. But over here, we have already calculated the change in momentum right? And so if we just divide by time, we are done, we'll get our force in one step. So it makes a lot of sense to use this equation. So again, can you try and give it a shot? See if you can use this equation and calculate what that net force is, which is the force of the wall on our clay. Go ahead. Pause the video and give this a try. Alright, if we plug in the change in momentum is negative 10 kilogram meters per second and divide by time that is .1 second, so that's it, so if we simplify we get our answer. So 10 divided by .1 is 100, so we'll get minus 100 and the units become kilogram meters divide by second and you have another second in the denominator which will get multiplied and become second square. So you get kilogram meters per second squared and by the way since this is the SI unit of force, this is also called Newton's, so let me just go ahead and write that as Newton's. And that's our answer. The force is minus hundred newtons. Okay and now let's look at what this negative sign is telling us. Well since the force is negative and the initial velocity is positive number, the negative sign is just saying the force must be opposite to the initial velocity. That makes a lot of sense. Right? The force must be in the opposite direction and that's why that clay ball is slowing down. Okay, let's do another problem which is similar to this but instead of a clay ball let's say it's a cricket ball. Now most of the stuff are the same, the only difference is that it hits the wall and it bounces back with the same speed, that's the main difference. So it doesn't stick to it but it bounces back with the same speed. So can you think of, you know how the situation would change now, maybe you need to draw a new diagram and what would now be the change in momentum and the force exerted by the wall? Again, pause the video and give this a shot first. Okay, so if we replace this clay ball with a ball of cricket the only difference we see now is that after hitting the wall it bounces back with the same speed, that means it's no longer at rest it will be coming back with the speed of 20 meters per second. So what will change? Well this problem is a little bit tricky and I'll tell you why. So when I should solve these problems earlier I would say "look, the only thing that has changed is that the final velocity is no longer zero". So I would write the final velocity is now 20 meters per second and as a result, the new change in momentum is zero. Why? Because look the final velocity, V is same as U and M is the same, that never change, so the final momentum is the same as the initial momentum that means change in momentum is zero. Right? That's the new answer. But that can't be true! Why? Because if the change in momentum is zero that means the new where it rule here becomes zero and that means the net force on that ball becomes zero. That means the wall is not pushing that ball at all, can that be possible? No! I'm pretty sure you agree with me when the wall goes and hits the wall, the wall must be putting a force on that particular ball. So clearly the force can not be zero, that means the momentum change can not be zero. So clearly something is wrong over here. Can you identify what's wrong over here? Again, pause the video and see if you can, figure this out. And this is super important and that's why, give this some time and think about, what is wrong over here. So did you get it? Did you see what was wrong? My big mistake was that I said the final velocity is the same as the initial velocity. That's not true, why? Because even though their speeds are the same remember velocity is speed with direction. And clearly the direction has changed, that means the velocity must have changed. But how do we represent that? Well since the final direction is the opposite of the initial direction, we can say the final velocity must be negative because the initial velocity is positive. That's how we represent opposite directions. And so, I must say the final velocity is negative 20 meters per second. That was the big mistake. So now that we have the correct values let's substitute this correct values and I'm pretty sure you can do these all by yourself. I want to save time over here and I'll just tell you what we end up with. So if we substitute the correct values, we'll not get zero now but we'll get negative 20 kilogram meters per second. If you just go ahead and check that you'll get negative 20. And so over here also when you divide that by .1 you'll eventually end up with negative 200 Newton. Again you can just clarify that. This will be the new answer. And again the negative sign is telling us that the force is in the opposite direction of the initial velocity, which again, kind of makes sense. And so remember since momentum is mass times velocity and velocity depends on direction be very careful with the direction. Whenever the direction becomes opposite remember to put a negative sign for that. And that will make sure that you don't do mistakes like I use to do before.