1. Quadratic Spline Interpolation Part 1 of 2 http://numericalmethods.eng.usf.edu

2. Quadratic Spline Example The upward velocity of a rocket is given as a function of time. Using quadratic splines a) Find the velocity at t=16 seconds b) Find the acceleration at t=16 seconds c) Find the distance covered between t=11 and t=16 seconds tv(t) sm/s 00 10227.04 15362.78 20517.35 22.5602.97 30901.67

3. Data and Plot tv(t) sm/s 00 10227.04 15362.78 20517.35 22.5602.97 30901.67

4. Solution Let us set up the equations

5. Each Spline Goes Through Two Consecutive Data Points

6. tv(t) sm/s 00 10227.04 15362.78 20517.35 22.5602.97 30901.67 Each Spline Goes Through Two Consecutive Data Points

7. Derivatives are Continuous at Interior Data Points

8. Derivatives are continuous at Interior Data Points At t=10 At t=15 At t=20 At t=22.5

9. Last Equation

10. Final Set of Equations

11. Coefficients of Spline iaiai bibi cici 1022.7040 20.88884.92888.88 3-0.135635.66-141.61 41.6048-33.956554.55 50.2088928.86-152.13

12. END http://numericalmethods.eng.usf.edu

13. Quadratic Spline Interpolation Part 2 of 2 http://numericalmethods.eng.usf.edu

14. Final Solution

15. Velocity at a Particular Point a) Velocity at t=16

16. Acceleration from Velocity Profile b) Acceleration at t=16

17. Acceleration from Velocity Profile The quadratic spline valid at t=16 is given by,

18. Distance from Velocity Profile c) Find the distance covered by the rocket from t=11s to t=16s.

19. Distance from Velocity Profile

20. END http://numericalmethods.eng.usf.edu