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Main points are: Differentiation-Discrete Functions, Forward Difference Approximation, Upward Velocity, Function of Time, Acceleration of Rocket, Direct Fit Polynomials, Data Points, Interpolant for Velocity, Matrix Form
Typology: Slides
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Example 1
The upward velocity of a rocket is given as a function of time in Table 1.
Using forward divided difference, find the acceleration of the rocket at.
t v(t) s m/s 0 0 10 227. 15 362. 20 517. 22.5 602. 30 901.
Table 1 Velocity as a function of time
1
1
To find the acceleration at , we need to choose the two values closest to , that also bracket to evaluate it. The two
points are and.
Solution
Direct Fit Polynomials
' n 1 ' x (^) 0 , y 0 , x 1 , y 1 , x 2 , y 2 , , xn , yn
nth Pn x a 0 a 1 x an 1 xn ^1 anxn
n ^ ^ n (^ ) a 1 ^2 a 2 x n^ ^1 a^ n 1 xn ^2 nanxn ^1 dx
dP x P x
In this method, given (^) data points one can fit a (^) order polynomial given by
To find the first derivative,
Similarly other derivatives can be found.
The upward velocity of a rocket is given as a function of time in Table 2.
Using the third order polynomial interpolant for velocity, find the acceleration of the rocket at.
t v(t) s m/s 0 0 10 227. 15 362. 20 517. 22.5 602. 30 901.
Table 2 Velocity as a function of time
Example 2-Direct Fit Polynomials cont.
such that
Writing the four equations in matrix form, we have
3 3
2
3 3
2
3 3
2
3 3
2
3
2
1
0
Example 2-Direct Fit Polynomials cont.
Solving the above four equations gives
a 1 21. 289
Hence
3 3
2 0 1 2
t t t t
v t a at a t a t
Example 2-Direct Fit Polynomials cont.
,
The acceleration at t=16 is given by
16 v t (^) t 16
Given that
v t
4. 3810 21. 289 t 0. 13065 t^2 0. 0054606 t^3 dt
d
21. 289 0. 26130 t 0. 016382 t^2 , 10 t 22. 5
In this method, given ^ x 1 (^) , y 1 ,^ ,^ xn , yn , one can fit a n 1 th order Lagrangian polynomial given by
n
i
0
n
j i
j (^) i j
j i x x
x x L x 0
( )
j i^ omitted.
2 2 0 2 1
1 1 0 1 2
0 0 1 0 2
2
2 2 2 f x x x x x
f x x x x x
f x x x x x
f x
2 0 2 1
1 0 1 1 0 1 2
0 0 2 0 1 0 2
2 2 1 2 2 2 f x x x x x
f x x x x x x x x
f x x x x x x x x
f x x x x
Differentiating again would give the second derivative as
Determine the value of the acceleration at using the second order Lagrangian polynomial interpolation for velocity.
t v(t) s m/s 0 0 10 227. 15 362. 20 517. 22.5 602. 30 901.
Table 3 Velocity as a function of time
The upward velocity of a rocket is given as a function of time in Table 3.