A radar records the range to a projectile with independent measurement noise of…

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Uncertainty in measurement

Error, Chance and the Machine, Ch Approximating a function: interpolation and differentiation

Problem

A radar records the range to a projectile with independent measurement noise of standard deviation σ=0.5 m\sigma = 0.5\,\mathrm{m} on each reading. Over the interval of interest the range's third time-derivative is estimated as ∣f′′′∣≈2 m s−3|f'''| \approx 2\,\mathrm{m\,s^{-3}}. A central difference is to be used to estimate the velocity from the range record. ((a)) Find the difference step h∗h^*, the half-width of the central difference [f(t+h)−f(t−h)]/2h[f(t+h)-f(t-h)]/2h, that balances truncation error against noise, that is, minimises the rms error when the truncation term h2∣f′′′∣/6h^2|f'''|/6 and the noise term σ/(2 h)\sigma/(\sqrt2\,h) add in quadrature. ((b)) Find the truncation term and the noise term at h∗h^*, and the resulting rms error in the velocity.

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