 Solution

ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2021/paper-1/section-c/c8/b/solution

Use

x = t

in picoseconds and

y = ln (R / R_{*})

. Since the time uncertainty is negligible, only the vertical errors enter. First-order propagation of uncertainty gives

σ_{y} ≃ \frac{d y}{d R} σ_{R} = \frac{σ _{R}}{R} = 0.15.

The transformed measurements are

x (ps) 0.25 0.75 1.25 1.75 2.25 2.75 y 7.19968 5.88610 4.70048 3.43399 2.63906 1.09861 σ_{y} 0.15 0.15 0.15 0.15 0.15 0.15

Thus the logarithmic transformation converts constant fractional rate uncertainties into approximately equal absolute log-rate uncertainties. The equality is the linear error approximation; finite error bars would become slightly asymmetric under an exact logarithm.

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