Impulse with Variable Forces

by Patrick Ford
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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