2. 1D Motion / Kinematics

# Calculating Change in Velocity from Acceleration-Time Graphs

1

concept

## Calculating Change in Velocity from Acceleration-Time Graphs

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Hey guys. So sometimes you'll have to calculate changes in velocity from the acceleration time graphs instead of using your equations. So that's we're gonna check out this video, but we're going to see that it is exactly like how he calculated displacements from velocity time graphs. Let's check it out. So, guys, when we had velocity time graphs the area under the curve, remember, was just the area that's enclosed between the values of the graph and the time access. You just make this little shape here and then your area under the curve represents your change in the position or the displacement. Delta X Well, for an acceleration time graph, it's the same exact principle. The area under the curve is just the area between the acceleration values and the time graph. Except now the area that's enclosed within the shape here doesn't represent the displacement. It represents your change in the velocity. So it's the same idea. It's just the variable that's different here. So that's really all there is to it, guys, it's the same exact procedures. We're just gonna be calculating a bunch of areas, so let's go ahead and check out this example. We've got an acceleration time graph for a moving box. We're told it's initially at rest. And so the party says we're gonna calculate the boxes velocity AT T equals three. So we know how to do this. You just figure out T equals zero t equals three, and then we just have to split this up and we have to calculate the area that's inside of this shape over here. This is not a very simple shape, and I don't know if the formula off the top of my head, but it can break it up into a rectangle or square and a triangle, and so I can calculate these areas should be Delta V one. I'm gonna call that and Delta V two and now I just have to figure out the total velocity by calculating and adding up these two things. Now, I'm also told in part a that it's not the change in velocity I'm looking for. It's the actual velocity. So what does that mean? What was asking for is V, um but I know have to calculate the velocity by first calculating the change in the velocity by looking at the area of the graph now remember that the change in the velocity Delta V is just the final minus v initial. And so and I'm told that initially the boxes at rest. So I know that this V not is just equal to zero. So the boxes velocity and the change in the velocity are the exact same thing. So that's really all it's asking me for. So we're looking for the velocity and I'm gonna call this V three, and all I have to do is just add up these two shapes together Delta V one in Delta V two. So Delta V. One is just a rectangle. It's just base times height, that's the area. Remember, the basis to and the height is too. So that means that's just gonna be two times two, which is four now. The units for this are gonna be in meters per second. Remember, because it is a change in the velocity. So you have to be careful. That's meters per second. Now Delta V two is just triangle, so it's gonna be one half base times height. That's the formula. The base of this is one. The height of this is too. So it's gonna be one half of one times two with one half and the two will cancel, and we'll just get 1 m per second. So now we just have to add these two sort of, like, smaller of values, and that's gonna be the total velocity of three seconds. So it's just gonna be four plus one, and that is 5 m per second. That's all there is to it. So now let's figure out what the velocity is. A T equals five. So all we have to do is now figure out the area under the curve for from 0 to seconds. And so that's gonna include everything from 0 to 3 plus now from 3 to 5. So that's all we have to do is we just have to tack on this extra area that we have between this little triangle here and the graph. So I'm gonna highlight this in green. So this is gonna be the area that I have to include now. It's not a very pretty shape. So what I can do is just break it up into two small right triangles. Here, I'll call this one Delta V four or sorry, Delta V. three. Let's call that. And then I'll call this one Delta V four. Um, but what you'll see is that they're symmetrical. They're the exact same pieces. So whatever I find for one value is gonna be the same for the other one. Okay, so the value or the velocity at five I'm gonna call that V five is just gonna be Delta V. It's actually going to be, uh, oops. It's gonna be the velocity that I found out in part A. It's gonna be the 5 m per second plus. And now it's gonna be Delta V three plus Delta V four. And so I already have what this is. I know this is 5 m per second, and now, plus, I could just add up together these other two velocities here, and this is gonna be my total velocity in meters per second. That's so that's the plan. So I've got Delta V three. I know this is a triangle, so I want to use one half base Times Heights. The base is one, and the height is too. So I'm gonna have one half of one times two, and that's again just getting to give me one. But again, remember that when you have areas that are above the time access, those are gonna be positive changes in your velocity because your acceleration is positive. Whereas if you have negative or so below the time access, then those were gonna be negative changes in Delta V because your acceleration is negative. So your velocity is becoming more negative. That's really all there is to it. So that means this is a negative one. And Delta V four is the same exact thing. It's also negative one. So this is gonna be negative one and negative one. And so your total velocity at five seconds is five minus two, which is going to be 3 m per second. And it's gonna be positive. So those are two answers. Let me know if you guys have any questions, let's get some more practice

2

example

## Sliding Block

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Hey, guys left Work this one out together, we've got an acceleration time graph. That's a versus T for a sliding block. And we're told that the initial velocity is 3 m per second and we're supposed to find is what's the blocks? Final velocity At T equals five seconds. So let's check it out. What is it that we're trying to find? We're actually trying to find what the final velocity is. And so this is gonna be our target variable. How do we find the final velocity? Well, first, we just need to figure out what the change in the velocity in general, the change in the velocity is just going to be my V final minus v initial. But remember that when it comes to acceleration time graphs the change in the velocity Delta V is actually just going to be the area that is under the acceleration time graph. And so we can rearrange this equation over here and figure this out. And so the final velocity or V is just gonna be the initial velocity plus the change in the velocity. In other words, just the area that's underneath the curve here. And so I already have the initial velocity that I'm given its 3 m per second. And so all I need to do is if I try to figure out the final velocity. I just need to figure out what's the change in V. So the words, what's the area that's underneath the curve? So let's go ahead and figure that out. Basically, I know that this is from T equals zero. Yes, that's my initial velocity. And I have to figure out that the final velocity T equals five. So that means that basically, I'm gonna be adding up all of the area that is between these two points on the curve and the area that that corresponds to underneath the graph is just everything that's gonna be included between the values and the time access. So it's gonna be all the stuff that's in green over here. Now again, this is weird, kind of like this triangle looking thing, but it's not really a nice shape that I need. So what I can do is break it up into more manageable shapes. Eso What I can do is I can draw line down here, and this will be one triangle and then I can draw a line over here, and that will be a rectangle and another triangle. So this should work. So now I'm gonna have three shapes over here. And so remember, I'm gonna have to find out what Delta V is. So let's write out an equation for this. My delta V is just gonna be If I have these three smaller shapes, it's just gonna be adding up all of the areas that are underneath those those shapes. So I'm gonna call this one Delta V one. I'll call this guy Delta V. Two, and I'll call this guy Delta V. Three, and you just have to use the right formulas for each one of them. So let's just get to it. Um, the total change in velocity is just gonna be once I add up all of the changes in velocity Delta V one, Delta V two, Delta V three. Okay, so let's get to it. So Delta V one is going to be a triangle, which means the area we're gonna use is one half base times height. So now I just have to figure out based on the height, So I've got 123 The base of the triangle is gonna be from 0 to 3. So that's three. The height is gonna be from two all the way up to six. So that means that the height of this triangle is four. So I've got one half of three times four, and so that's gonna be 12. 1 half of 12 is six. And remember, this is gonna be in meters per second. So Delta V two is just gonna be a rectangle. Eso you just use base times height for this and the base of this triangle or sorry, this rectangles gonna be three. The height of this is gonna be too. And so we just have three times to and it's also 6 m per second. And then now, finally, let me just rewrite this actually, eso that's adult of you three. So I've got one half of base times height and so the base of this triangle is gonna be to the height of this triangle is actually gonna be six, because going from zero all the way up to six here. So just make sure you have your bases in your heights. Um, sort of worked out so we've got one half of base, which is to height is six. One half of the two will cancel and then we'll just end up with 6 m per second. So this is kind of interesting. Notice how we get 6 m per second for all the shapes and six, even though they all kind of look different, obviously, all your numbers won't work out toe, Always six. You have to, you know, do the do the math for each one of them. But it's kind of interesting how all these different looking shapes we had ended up getting the same area. So we actually have all of these three things here, which means we can calculate the total change in the velocity Delta V just six plus six plus six, which equals 18 m per second. So now are we done? Is this our answer? Well, no, Remember, this represents the change in the velocity, not the final velocity. So we actually have to plug this back into this equation over here to figure out the final velocity. So let's go and do that. Our final velocity is gonna be our initial velocity of 3 m per second. So it's gonna be three plus now the change. And now the changes 18. So we get a final velocity of 21 m per second and this is our final answer. Alright, guys, let me know if you have any questions. Let's keep you