Compute the mass m and its error by error propagation
Covariance and Correlation
We denote the mass and the velocity of an object by m and v and their measurement errors by
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The measurements are assumed to be independent, i.e., cov (m, v) = 0. Furthermore, the relative errors of measurement are known, i.e.,
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(a) Consider the momentum p = mv and the kinetic energy E = mv2of the object and compute σ2 (p), σ2 (p), cov (p, E), and the correlation p (p, E). Discuss p (p, E) for the special cases a = 0 and b = 0. Hint: Form the vectors x = (m, v) and y = (p, E). Then approximate y = y(x) by a linear transformation and finally compute the covariance matrix.
(b) For the case where the measured values of E, p and the covariance matrix are known, compute the mass m and its error by error propagation. Use the results from (a) to verify your result. Note that you will obtain the correct result only if cov (p, E) is taken into account in the error propagation.
