A sonographer scans a carotid artery with a probe

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

Life at Thermal Energy, Ch Echoes, precession and the therapeutic beam

Problem

A sonographer scans a carotid artery with a 4 MHz4\,\mathrm{MHz} probe. The returned Doppler shift is measured very precisely as Δf=150 Hz\Delta f = 150\,\mathrm{Hz}. The beam-to-flow angle cursor is set to θ=87∘\theta = 87^\circ, but the operator can only align the cursor with the vessel wall to within ±3∘\pm 3^\circ. Take the speed of sound in soft tissue as cs=1540 m s−1c_s = 1540\,\mathrm{m\,s^{-1}}. ((a)) Using θ=87∘\theta = 87^\circ, calculate the blood speed vv from Δf=2fvcos⁡θ/cs\Delta f = 2fv\cos\theta/c_s. ((b)) Recalculate vv for the two extreme angles the ±3∘\pm3^\circ uncertainty allows, θ=84∘\theta = 84^\circ and θ=90∘\theta = 90^\circ. ((c)) The sonographer reports the answer from part (a) as a reliable measurement, reasoning that Δf\Delta f itself was measured to three significant figures. Explain why this reasoning is wrong.

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