FIGURE EX2.31 shows the acceleration-versus-time graph of a particle moving along the x-axis. Its initial velocity is v0x = 8.0 m/s at t0 = 0 s. What is the particle’s velocity at t = 4.0s? <IMAGE>

Video transcript

Hey, everyone in this problem, a car is initially moving at a velocity of 5 m per second. It's acceleration versus time graph is given below. And we're asked to determine its velocity at T equals eight seconds. We have our graph. We're gonna look at that in just a minute. But we have four answer choices in this problem all in meters per second. Option A 18, option B 23 option C 46 and option D 51. So let's think about what we have the graph we're given is an acceleration versus time graph. So we have the time in seconds on the X axis, the acceleration of meters per second squared on the Y axis. And we wanna find a particular velocity. Now recall that the acceleration A is the derivative of the velocity with respect to time DV by DT. So if we reverse that, then V is gonna be the integral of the velocity, sorry, the integral of the acceleration. OK. So another way to think about this, if we're given a graph, how can we relate that graph of the acceleration to our velocity? Well, it's just gonna be the area under the curve. And so when we look at this graph that we have, we have this sloped section starting in the, on the left side moving upwards. And then we have this horizontal section and we can break it down into three different areas. So we have a triangle. I'll write that in green, that'll be area one. We have a rectangle that forms area two on the right hand side. And at the bottom, we have this big long rectangle that forms area three and we have to go all the way down to that horizontal axis. So we can't forget to include that third area. And so the change in volume in this case will be the area under our acceleration versus time curve. OK? Now that acceleration or that area is gonna be made up of those three components, a one plus a two plus a three. Now the change in velocity. Well, what we can see is that we're going from zero seconds all the way over to, well, our graph goes to 10 seconds, but we wanna go to eight seconds. We want that final the duration to be eight seconds. So let's cut off the area that we are gonna look at to that eight second time. All right. So that change in velocity is gonna be the final velocity, which is a velocity at eight seconds minus the initial velocity or the velocity at zero seconds. OK? The area a one that's the area of a triangle. So that is gonna be one half multiplied by the base multiplied by the height. Our next area is a rectangle. So that's gonna be length multiplied by width. And then we have another rectangle. So length two multiplied by width too. We can go ahead and substitute in all of these values. We have the velocity at eight seconds minus the velocity at zero seconds. Well, the velocity at zero seconds we're told is 5 m per second. So minus five meters per second is equal to one half, multiplied by the base of our triangle. Well, it goes from zero seconds up to six seconds. So that's gonna be six seconds. The height we go from 2 m per second squared to 8 m per second squared. That's a change of 6 m per second. So multiplied by 6 m per second squared. Then we add the length times width of our rectangle A two and we can see that we go from six seconds up to eight seconds. So that's gonna be two seconds and then we go from 2 m per second squared up to 8 m per second squared. That's gonna be 6 m per second. Ok? So two seconds multiplied by 6 m per second squared. And finally, we add this final area, ok. This rectangle, we're going from zero seconds up to eight seconds. So we have eight seconds and then we're going from zero up to 2 m per second squared. So we multiply by 2 m per second squared. When we work this out, we get that the velocity at eight seconds is going to be equal to 18 m per se who was 12 m per second plus 16 m per second. Pull us 5 m per second. So we have the area of our three sections and then we have that initial velocity. It was on the left hand side, we've moved it to the right hand side and we can simplify everything on the right hand side, we can add it all together and we'll find that the velocity at eight seconds is going to be equal to 51 m per second. Ok. So what we did is we used this idea that the change in velocity is the area under an acceleration time curve. We calculated the areas and we were able to find that the velocity at eight seconds is 51 m per second which corresponds with answer choice. D Thanks everyone for watching. I hope this video helped see you in the next one.