9.2 Impulse Graphs

Today, we're going to look at impulse graphs. Remember, impulse is the change in momentum, so we're going to analyze what happens when we represent this on a graph and what we can infer from it. But before we do that, let's quickly review. Remember, we have two formulas for impulse: force multiplied by time or mass multiplied by velocity. We need to remember what is constant and what is changing. In the right-hand form, the mass is constant. Typically, the mass of an object does not change, but the velocity can. That's why we have the symbol ΔV in the formula. On the left-hand side, we have ΔT, which represents the change in time from the start to the finish. Time itself may not actually be changing, but we measure the difference in time between the two points. Therefore, using the same logic, force must be a constant. If force is constant in the formula, it means that the amount of force remains the same throughout the entire time period. This also implies that constant acceleration. So, the key thing to remember then is that when we're using our impulse formula (force times time), that is only for a constant force. If the force isn't constant, then we can't use that formula. So let's look at the situation where it is constant and see what happens when we graph it. If we were to look at a graph or an easy one, we would understand that it'll help us solve questions when forces won't be constant. So, let's start with a basic graph where our force is going to be constant throughout a period of time. A constant force will have just a straight horizontal line. So, if we were to calculate the slope of that line (rise over run), the rise would be our force and the run is our time period. In this case, we know that our slope is 0, so really this slope doesn't mean anything. So, for a normal kind of equation where the slope would be slanted, we'd egregiously rise over run, which would then be force over time. So, with this kind of graph being in constant force, the graph really doesn't help us much. But what we know from even in physics 20 when we did this is, if we look at, let me just kind of redraw the graph. So, if we have our force again being constant and we have a certain time that we were examining, now what we can do is we can say, well, the slope doesn't mean anything, but the area under the curve does. So, let's look at sort of what that area under the curve would look like. Okay, so we've got this entire rectangle. So, if we look at the area under the curve, the area formula would be base times height, which would then just be our change in time times our force, the force times... Time, so we can see that from that equation we just get basically force times time, which would then be our area of the graph. This is actually our impulse because we know from our formula (my formula is messed up here) that impulse is Force times time, right? So, if we go back to the graph and let's do this first example: if we have a 25 Newton force over 10 seconds, we can see that our area under the curve would be base times height, which would be 25 times 10, equaling 250. But we also know from the formula that impulse is Force times time. Time, so that would give us 25 times 10 as well because it's constant. So we get the same answer. The key thing is that we can see that when it's constant, we may as well just use the formula. We'll just go 25 times 10. But if it's not constant, we see that the area will give us the same answer. So when we do get questions where they're not constant, that's when the graphs will come in handy. We cannot use the formula, right? The formula is only used for constants. So if it's not constant, then we do have to look at the graph. And that's really our only option. So let's do a couple of examples where the force isn't constant. This first one says we have a car going for breasts, and it's gonna accelerate over 5.78 seconds. And we're gonna increase our force from 0 up to 30 12 Newtons. So if it's a steadily increasing kind of graph, we should get just a straight diagonal line, kind of like that. So our force is gonna go up to 2012 and our time is going up to 5.78 seconds, and the mass of the car is 1500 in case we need that. So the question is, what is the impulse and what is the final velocity? Okay, so what we want to do with this one is find the impulse as the area under the curve because it's not constant, we're gonna have to find the area. In this case, our area will be the area of this triangle. The area of a triangle is calculated as base times height divided by two. So we would have a base of 5.78 times our height of 2012, and divided by two. That gives us an area of 8700.4. Seven, so we round that off to eight point seven zero times ten to the three kilograms meters per second. So that's our impulse. Okay, so that part's pretty straightforward. Now, the question is, how do we get our final velocity? Our final velocity is going to be determined using other formulas. Let's do that over here on the side. Remember, we know that impulse is equal to force times time (if it's a constant force), but we also know that it's equal to mass times change in velocity. We already figured out that the impulse is eight point seven times ten to the three. The mass given in the question is fifteen hundred kilograms. So, we can solve for velocity using this information. Our change in velocity is assumed to be starting from rest, as stated in the question. So, essentially, we're just looking for our final velocity, which would be the difference. So RBF VI. VI is zero, we're just really looking for VF. So our VF then would be 8.73 divided by 1500, and we'll get an answer of 5.80. And that's it! So a couple of key things to remember: our force times time formula only works for constant forces. If it's not constant, we're going to have to use a graph and figure out the area. And then after that, we can still use our mass times velocity. In any case, let's do another one. Now, this one is not a nice, easy right-angle triangle, but it'll still work the same way. In this case, it says we've got a graph showing 500 kilograms meters per second on the y-axis and time in seconds on the x-axis. We can see from the graph that for the first 2 seconds, there's a constant force of 250 newtons. Then, from 2 to 4 seconds, there's no force acting. And finally, from 4 to 7 seconds, there's a constant force of 150 newtons. We're asked to find the impulse during this time interval. To find the impulse, we need to find the area under the force-time graph. We'll break it up into three parts: the first rectangle with a length of 2 seconds and a height of 250 newtons, the triangle with a base of 2 seconds and a height of -250 newtons, and the second rectangle with a length of 3 seconds and a height of 150 newtons. For the first rectangle, the area is 250 newtons times 2 seconds, which is 500 kg m/s. For the triangle, the area is 1/2 times base times height, or 1/2 times 2 seconds times -250 newtons, which is -250 kg m/s. For the second rectangle, the area is 150 newtons times 3 seconds, which is 450 kg m/s. Now we just add up the areas: 500 kg m/s plus -250 kg m/s plus 450 kg m/s, which gives us a total of 700 kg m/s. And that's the impulse during this time interval. For our next example, the question is asking us to find the maximum force. It tells us that the impulse is 500 kg m/s and the time is 10 seconds. In this case, since the impulse is equal to the base times height divided by 2, we can rearrange the equation and solve for the height, which represents the maximum force. So we have (base times height) divided by 2 equals 500. We know the base is 10, so we need to find the height. Dividing both sides of the equation by 10, we get height equals 100. Therefore, the maximum force is 100 Newtons. In summary, for any kind of graph, as long as we know that the area is the impulse, we're in good shape. When the force is constant, we can use the force times time formula. But when the force is not constant, we need to use graphs to find the area. And if the graph is a curve, we can approximate the area using triangles or other shapes. That's all for today.