 Okay, so I just made up a problem here that we can use forces and solve. Let's just see how it goes. So suppose I have some ball, like I said it didn't give its mass. It has a mass of 0.5 kilograms. And I shoot it, I kick it, something, I don't know, 0.5. It could be a golf ball. But I launch it at 4 meters per second at an angle 30 degrees above the horizon. And I want to know how high it goes. So let's call this H. That's what I want to find out first. Okay, so when we have a problem like this, we want to use the momentum principle. So to use the momentum principle we need to know the net forces. So after this thing's kicked, what forces are acting on that ball? Well, think about when you're drawing a force diagram like this, think about what things are touching the ball and what long-range forces are acting on the ball. So far the only long-range force you've seen is gravity. So everything else has to be touching the ball. So if you wanted to say air is touching the ball, that's a pretty valid assumption. But in this case, I'll tell you that 4 meters per second, that's probably a small enough force to ignore the air resistance. The gravitational force is not. So we have this. That's the only force that's acting on the ball. There's no force from the kick or the throw or anything like that. Because the foot's not touching the ball after it leaves. So it's just that. That's the only force. Okay, the whole time it's going up and back down. It's just this constant force because the earth doesn't change. Okay, so what can we say at the top, the highest point? If it's going up, it's slowing down and goes up, right? Because the force is pulling down. And eventually it reaches some highest point. And then it starts to come back down. So at the highest point, the vertical velocity is zero. So if I call this the x-y direction, then here, v-y is zero meters per second. And I want to find out how high it is when it gets there. Let's do something. Let's just write down the momentum principle. Okay, so the momentum principle says f-net equals delta p over delta t. So do I know the net force? Well, yeah, I do. It's just mg. Do I know the change in momentum? Kind of. Do I know the change in time? No. But that's okay. Let's just go ahead and put in what we know. So first, look at the initial momentum. If we'll call this p1, if we'll call that p2 up there. So if I say, if I want to write p1, it's going to be 0.5 kilograms times initial velocity. So in this case, that's going to be 4 times the cosine of 30. 4 times the sine of 30 is zero meters per second. So if I look at this, if you have some vector like that, here's the x component of velocity and that's 30 degrees. That's 4. The hypotenuse is 4 meters per second. So the adjacent side, which is the x component, is going to be the hypotenuse times the cosine of 30 and the y component is going to be the hypotenuse times the sine of 30. That's how I get that. Okay, what about the momentum at the top, p2? Well, p is still the same mass, 0.5 kilograms. What about the velocity? Well, I know that the force is only in the y direction. So this is only going to be equal to, let me go ahead and write down that force, f net, the zero, negative mg, zero, and when I write g as a scalar, I'm saying it's 9.8. So since the x and the z components of forces are zero, the x and z components of the change momentum have to be zero. So that means that the x component doesn't change. 4 times cosine 30. And then I already said, I know what the y component was. It was zero. So now I know p1 and p2. So now I can actually solve for delta t. Let me just look at the y component of this vector equation. That says negative mg equals p2 minus p1 in the y direction. So it's going to be zero minus 4 sine 30. I'm going to leave it off the units delta t. Let me solve for delta t. Delta t is going to be 4 sine 30 degrees over mg. Let's just check to make sure it has the right units. This is in meters per second. And this is in newtons per kilogram. So I get meters per second over newtons. But a newton is a kilogram meter per second. Where the kilograms go? Newtons is a kilogram meter per second squared. That's in newtons. Momentum. See, it's a good thing to check your units. I put the velocity. I didn't put the momentum. So the mass is there. Okay. This is momentum. I put 4 sine 30 is the velocity, not the momentum. See, this is why you check your units. So this is going to be tons. The mass. So now I get meters per second squared. Meters per second divided by meters per second squared does give me units of seconds. So the mass cancels. So it's just going to be 4 sine 30 over 9.8 seconds. And I'm going to leave it like that. I'm going to put it in your calculator and you can do that. Okay, now you should say, but wait. That's not what we wanted. We wanted how high it goes. So let's look at the y-direction again. Just the y-direction because that's all we care about. And let's say this is y equals 0. Y1 equals 0 meters and y2 equals h. So in the y-direction I can use this, the y-average. This is just a y-direction so I'm not writing it as a vector. It's going to be the change in r, y, the y position over the change in time. Well, I have the change in time. Okay. So I can put that in there. And I want to find the change in r. So I just need this v-average. Well, since the velocity is changing linearly with time, that means constant acceleration, which is when you have constant forces, then v-average, vy-average, is going to be vy2 plus vy1 over 2. You're just average of velocities. So we already said at the highest point the velocity is 0. This is 0 plus 4 sine 30 over 2. So now I can use this to solve, I can write this as final is going to be h minus the initial y-position is 0 over delta t. So now I get h equals v-average, which is just going to be 4 sine 30 over 2 times delta t. But delta t is 4 sine 30 over 9.8. And there's my height. So let's just check the units. You always want to do that. This 2 is just a 2 from an average, so it has no units. So this is in meters per second. We already said that's in seconds. So I get meters per second times seconds does give me meters. Okay, let's just put this in the calculator just so you can get a value because otherwise people say you never put in numbers. Let me choose my calculator right here on my computer. It doesn't look right. I get 0.2. Oh yeah, so I think that's okay. That's not very high, right? Something like that. It's not going very fast. And it's going at a very, you know, not shooting it straight up. So the vertical component of loss is pretty small. Okay, but let's just look at the key points here. In this case, I use the momentum principle, which I deleted. I use the momentum principle over here. This is the momentum principle. And I use that to find the change in time. You know, remember that that's what it gives you. But I wasn't looking for the time. So I use this average velocity and the position update formula to get the height. And that's what it did. Okay, hopefully that helps.