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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1/12/d/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 12 / d / Solution by

Codex  0 2026-10-18

Rewrite the complex position as

z (t) = 2 + 3 i / (t - i)

. Differentiating gives the complex velocity and acceleration

\overset{z}{˙} = - \frac{3 i}{( t - i ) ^{2}}, \overset{z}{¨} = \frac{6 i}{( t - i ) ^{3}} .

(1)

For real

t

∣ t - i ∣ = t^{2} + 1

. Their requested magnitudes are therefore

∣ \overset{z}{˙} ∣ = \frac{3}{t ^{2} + 1}, ∣ \overset{z}{¨} ∣ = \frac{6}{( t ^{2} + 1 ) ^{3/2}} .

(2)

There is no pole for real time. The magnitude is the Euclidean length of the two Cartesian components, not the modulus of either component separately.

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