Question

1. A player hits a baseball into the air. A motion detector is used to gather data about the height of the ball. The resulting data is recorded in the table. Find the finite differences. Decide whether the data can be modelled by a linear, a quadratic, a cubic, or a quartic function. Then use the appropriate regression to determine the algebraic model for the data. DATA TABLE Height (m) First Differences Second Differences Third Differences Time (s) 0 Fourth Differences 1 2 3 4 5 6 7 0.5 30.6 50.9 61.4 62.1 53.0 34.1 5.4 What model is the best fit for this data? How do you know?? Equation for the curve of best fit:

Answer 1

Given: Time , X 0 3 4 j 6 7 30.650.9|61-4/62-1/53-934-15. Height 0.5 y time Here, as the difference between intervals is same, any interpola bon method either Lagrange's (or) Newton's will be best fit a) Newton's interpolation model is the best fit for this data, b) Because the difference between two consecutive intervals is same re; equal height c) Newton's forward difference table is, first second Third Fourts diff (44) drff (A?y diff (o'r) diff (194) х У 0.5 30.1 I 130,6 - 9.8 20. 3 2 50.9 9.8 o 10.5 o 3 61.4 -9.8 o 0,7 -9.8 0 0 4 62.1 o -9.1 5.153 -9.8 0 -18.9 34-1 - 9.8 - 28,7 7:15.4

inteopolation Newton's forward difference formula is, y(x) = Yo + P s Yo + 3. yo x-xo X-O X. P- h these values, we substituting get, 4(x)= 0.5 +(x * 30.1) + x (x-1) x (-928) 2 *(x-1) (x-2) + 6 -0.5 + 30.1 x + (-4,9) (x2-x) 4.9x?+ 4.9% = O, S + 3o, ) x Y(x) = -4.9x2 +358 +0.5 by the Hence, given data can be modelled function quadratic and the equation, ly(x)= -4.9*?+35* +0.5 the is of curve the best fit. of the equation. Where, Х -> time. 4 -> height:

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