Hey guys, let's 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 meters per second and we're supposed to find what's the block's final velocity at t equals 5 seconds. So let's check it out. What is it that we're trying to find? Well, we're actually trying to find what the final velocity is and so this is going to be our target variable. How do we find the final velocity? Well, first, we just need to figure out what's the change in the velocity in general. The change in the velocity is just going to be my v final minus v initial. 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 going to 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. It's 3 meters 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 in other 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 0. Okay. So that's my initial velocity and I have to figure out the final velocity, t equals 5. So that means that basically I'm going to 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 going to be included between the values and the time axis. So it's going to be all the stuff that's in green over here. Now again, this is weird. It's 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. So what I can do is I can draw a line down here, and this will be one triangle. And then I can draw a line over here, and that'll be a rectangle and another triangle. So this should work. So now I'm going to have 3 shapes over here. And so remember, I'm going to have to find out what delta V is, so let's write out an equation for this. My delta v is just going to be if I have these 3 smaller shapes, it's just going to be adding up all of the areas that are underneath those, those shapes. So I'm going to call this one delta v one, I'll call this guy delta v two, and I'll call this guy delta v 3. And you just have to use the right formulas for each one of them. So let's just get to it. The total change in the velocity is just going to be once I add up all of the changes in velocity. Delta v 1, delta v 2, delta v 3. Okay. So let's get to it. So delta v one is going to be a triangle, which means the area we're going to use is math>1/2 base times height. So now, I just have to figure out the base and the height. So I've got 1, 2, 3. The base of the triangle is going to be from 0 to 3, so that's 3. The height is going to be from 2 all the way up to 6. So that means that the height of this triangle is 4. So I've got 1/2 of 3 times 4, and so that's going to be 12. 1/2 of 12 is 6. And remember, these are going to be in meters per second. So delta v 2 is just going to be a rectangle. Rectangle, so you just use base times height for this, and the base of this triangle, or sorry, this rectangle's going to be 3, the height of this is going to be 2, and so_we just have 3 times 2, and that's also 6 meters per second. And then, now finally, let me just rewrite this actually, so that's delta v 3. So I've got 1/2 of base times height, and so, the base of this triangle is going to be 2. The height of this triangle is actually going to be 6, because we're going from 0 all the way up to 6 here. So just make sure you have your bases and your heights, sort of worked out. So we've got 1/2 of base, which is 2, height is 6. 1/2 and the 2 will cancel, and then we'll just end up with 6 meters per second. So this is kind of interesting. Notice how we get 6 meters per second for all the shapes. 6, 6 and 6, even though they all kinda look different. Obviously, all your numbers won't work out to always 6. You have to, you know, do the do the math for each one of them. But it's kinda interesting how all these different looking shapes, we ended up 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, it's just 6 plus 6 plus 6, which equals 18 meters 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 ahead and do that. Our final velocity is going to be our initial velocity of 3 meters per second. So it's going to be 3 plus now the change, and now_pain 18. So_blog_ final velocity of 21 meters per second, and this is our final answer. Alright, guys. Let me know if you have any questions. Let's keep
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2. 1D Motion / Kinematics
Calculating Change in Velocity from Acceleration-Time Graphs
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