Impulse with Variable Forces

Hey guys, so in some problems are going to be given force versus time graphs, like this graph over here, and you're gonna be asked to calculate the impulse. So I'm gonna show you how to calculate impulse by using these graphs here. But the idea is gonna be very straightforward, it's gonna be very similar to something we've already seen before. Now when we talked about work, we said that one way to calculate work was if you're given an F vs. X graph, the area under the graph was going to be equal to the work. And we said that areas above the X axis, we're going to be positive work. So if you have an area like this, this is going to be positive work because your forces positive here on the side of the axis, areas below the X axis were negative works because the force is negative. It's the same idea here when you have F versus T graphs, right? Except instead of calculating the area under this craft and that being the work the area underneath these efforts to graphs is really just going to be the impulse. Alright. But the rule is actually gonna be the same, right? So areas above the X axis are actually gonna be positive impulses because the force is going to be positive, Right? So you're positive force, positive impulse and the areas below are just gonna be negative impulses like this. So, the idea is very similar. It's really the letter that's different versus delta T. All right. So, let's take a look at an example here. So, we have a remote controlled car that is moving forwards and backwards. And it's sort of due to this changing force here that we have. So we want to calculate the impulse that's delivered to the toy car. So, remember that your momentum, or sorry, that your impulse in part A is going to be F. Times delta T. Except we can't really use this formula because this force constantly changing over time. So we say here, this is actually just equal to the area that's underneath this curve right here. So what I'm gonna do is say that the air that the impulse, I'm just basically going to break up this graph into two different sections here. So I'm gonna call This guy right here, the area from 2-4. I'm going to call this a one and this area over here from 4 to 9 seconds. Right? So this is like this is T equals nine. Here is I'm going to call this A two. And really to calculate the impulse. All I have to do is just add together these areas here. Break some stuff up and do a bunch of triangles and rectangles. And then just, you know, just add everything up together. All right, so let's check this out. So the area one. So really if I sort of break this up using this triangle right here, using this line, this is gonna be a triangle. So the base is equal to two and the force is equal to 10. So, remember that the area of a triangle is one half base times height. So this is gonna be one half of two times 10. And that this is gonna be the area of this guy right here, which is this area of the rectangle, which is just the base which is to and then the area and the height which is 10. So, if you go ahead and work this out, what you're gonna get, you're gonna get 30 you're gonna get 30 right here. So now we'll talk we'll calculate a two. So a two is gonna be basically all the negative stuff, right? All the areas below the X axis. So what I can do is I can break this up into a triangle, then this is a rectangle and then this is going to be another sort of triangle. So the area two is going to be the area of this rectangle. Here is gonna be one half the base is too, and the height is actually negative five here, that's the force. So this is gonna be two times negative five. And this is gonna be plus. Now I've got this area right here which is the base of two and a height of negative five. Again, remember always put the negative science because that's what's gonna give you negative impulses. And then this guy right here is gonna be one half with this is actually equal to one because we're going from 8 to 9. So this is gonna be one half of one um times this is gonna be negative five. All right, So, let's see what do you get here? You actually get 10? You're gonna get negative 10 here, that's going to be the area. So to calculate the impulse, you're just gonna do 30 plus negative 10 and you get a total impulse of 20. So it's gonna be 20 newton seconds here, and that's your answer. So, now what about part B and part B. Now we have to calculate the final speed of the toy car. So what this means is that this toy car is going to be traveling at some final speed. As we want to calculate, we're told that the toy car initially starts from rest, so the initial is equal to zero. How do we do that? We're just gonna go back to our impulse equation. Remember this is the impulse equation. We write that J. Which is equal to f times delta T. Is equal to M. V. Final minus M. V. Initial. Now we didn't really calculate F delta T. Because we didn't actually use that equation, but we really did calculate this because we calculated the area that was underneath this curve and that's basically what the impulse was. So we know that this impulse here was equal to 20 and this is just going to be equal to the final minutes, mp initial. There is no initial momentum because the initial velocity is zero. So we just right That the mass, which is two kg, it's gonna be over here times V final, which you should get is that the final is equal to 10 m per second. Simple as that. So that's what this one guys

Hey everybody. So in this problem we have a baseball bat that's hitting a bat and we have this graph here that shows the force over time. So it's a massive amount of force. But it happens with a very very small amount of time. The scale is in milliseconds here. Let's jump into the first part here. We want to calculate the impulse that is delivered to the baseball. So in other words we want to calculate is j remember j whenever you're given a force versus time graph is it's going to be the area that is under the curve. So in other words, you just take this graph here, you split this up and do a bunch of triangles and squares and stuff like that. And you just calculate the area that's under the curve. So this shape is kind of like a triangle but it's kind of weird because this side here is a little bit different than the right side. But what we can do is we can break it up into a smaller to smaller right triangles. And I'm gonna call this one a one and a two. So to calculate the impulse, we're gonna calculate the area and that's really just the area of a one plus a two. All right, so let's go ahead and write out these formulas here. Um Well a one is gonna be this has this is a base, I'm gonna call this base one and this has a height of h the other triangle has a base. I'm gonna call base to and it also has the same height. So it's the H. This is the same age for both of the triangles. So that means that the area formula for area one is going to be one half of base one times height plus one half of base two times heights. And what we can do here is we can just do a little math or geometry trick, which is we're gonna combine these two bases together. This is gonna be one half, base one plus base two times height. So really all you need to calculate for any triangle doesn't necessarily have to be a right triangle is you need to base the whole entire thing times the heights and then you multiply it by one half. Alright, that's just a cool little shortcut. But anyways so we're actually going ahead and plug this in. So we've got the base one which is gonna be what are we gonna plug in four. Well remember that this scale here is in milliseconds and we have to convert it to the right units. Four milliseconds is really just 0.4. So be careful here because this four here is going to be 0. seconds. Likewise, what happens here is we have to do the base too. This is going to be 8:00 which is going to be 0.008. So we add that in 0.008. And then you have to multiply by the height of the whole triangle which is 115 100 when you work this out, what you're gonna get is an impulse of nine newton seconds. Alright. So even though the force is massive it's 1500 it acts over a very small amount of time. And so you end up with something you know like nine which is a super huge number. Alright, so that's the answer. That's the impulse. Let's move on to the second part here. Now we want to calculate the final speed of the baseball right after the impulse, right? So after the bat hits the ball and it goes off, we want to calculate what is the speed after that impulse is delivered? So how do we do that? We'll remember that the impulse can only always be related to the change in the momentum. Another formula that we have here is that J. Is equal to force times time, but it's also equal to basically the change in the momentum. So we have M. V final minus M. V. Initial. What we really want to calculate here is the final. So if you look through what happens is that the second term, the mv initial, this is the initial momentum is going to be zero. What happens here is that there is no velocity, there's no initial velocity because the baseball is initially at rest, right? So there's no initial speed here. So the whole entire term goes away. So what's the J. Well we just calculated it, it's the nine newton seconds. So this is gonna be nine is equal to and then we've got the mass. The mass is 200 g. So this is gonna be 0.2 kg. This is gonna be zero point to the final. So if you work this out, what you're gonna get here is nine divided by 0.2. And that's the final speed of 45 m per second. And that's pretty reasonable because this is about, you know, let's say 100 miles an hour and that's pretty reasonable for a baseball, right? You can get, you know, you hit the baseball and it goes off like 100 miles an hour. That's someone reasonable. Alright, So that's the answer. Let me know if you have any questions.