A motion sensor records the height (in metres) of a bouncing ball's centre of mass

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Computational Physics, Ch Differentiating and integrating numbers

Problem

A motion sensor records the height

y(t)

(in metres) of a bouncing ball's centre of mass, which over one bounce is modelled as

y(t) = t^4

(an idealised model used only for this calculation, in SI units). Five readings are taken at

t = 0.8, 0.9, 1.0, 1.1, 1.2\,\mathrm{s}

, giving

y = [0.4096, 0.6561, 1.0000, 1.4641, 2.0736]\,\mathrm{m}

. (a) Form the second difference of this array by offset slicing with

\Delta t = 0.1\,\mathrm{s}

, and state the length of the resulting array. Give its value at

t=1.0\,\mathrm{s}

, compare it with the exact acceleration

\ddot y(1.0)=12t^2\big|_{t=1}

, and check the difference against the predicted leading error term

\tfrac{1}{12}\Delta t^2\,y^{(4)}(1.0)

. (b) Central-difference estimates of the velocity

\dot y(1.0\,\mathrm{s})

from the same record give

D_{0.2} = 4.16\,\mathrm{m\,s^{-1}}

and

D_{0.1} = 4.04\,\mathrm{m\,s^{-1}}

. Combine them by Richardson extrapolation and compare with the exact velocity

\dot y(1.0)=4t^3\big|_{t=1}

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