CHAPTER 05.04: LAGRANGE METHOD: Quadratic Interpolation: Example: Part 2 of 2
So in order to calculate v of 16, it'll be equal to 16 minus 15, t minus t1, divided by t minus t2, divided by t0, which is 10, minus t1, times t0 minus t2, that's 10 minus 20, multiplied by the velocity at t0, which is 227.04. So the second part is, again, t minus t0, which is 16 minus 10, times t minus t2, which is 16 minus 20, divided by 15 minus 10, which is t1 minus t0, and then you have t1 minus t2, which is 15 minus 20, times the velocity at t1, which is 362.78. And then we have the third part, which is t minus t0, which is, again, 16 minus 10, times 16 minus 20, which is t minus t2, divided by t2 minus t0, which is 15 minus 10, and then t2 minus t1, which is 15 minus 20, times the velocity at t2, which is 362.78. Sorry, it's 16 minus 15, so this is not right. Let me just rewrite this third part, it will be 16 minus 10, t minus t0, times t minus t1, which is 16 minus 15, divided by t2 minus t0, which is 20 minus 10, t2 minus t1, which is 20 minus 15, times v of t2, which is 517.35. So those are the Lagrangian weight which you are getting for the three different velocities which you have. So what I want to do is rather than calculate the whole number for you, I'm going to show you what each of these get weighted by. You have -0.08 times 227.04, this is the weight which is given to the velocity at t0, plus 0.96 times 362.78, so that's the amount of weight which is given to the velocity at t1, plus 0.12 times the velocity at t2, which is 517.35. So you can very well see that the amount of weight which is given to each of these velocities at t0, t1, t2 is -0.08 here, 0.96, and 0.12 here. If you add all these weights, it will turn out to be 1, of course, so that' s the way you should interpret Lagrangian interpolation. So if you do this, we get 392.19 meters per second. So that's what turns out to be the velocity at 16. And that's the end of this segment. |