 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ia/paper-4/12a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

m (\ddot{x} + 2 ω \times \dot{x} + ω \times (ω \times x)) = - 9 m ∣ ω ∣^{2} x .

For

ω = (0, 0, ω)

and

ξ = x + i y

, the planar equations combine to

\ddot{ξ} + 2 iω \dot{ξ} + 8 ω^{2} ξ = 0.

ξ (t) = \frac{2}{3} e^{2 iω t} + \frac{1}{3} e^{- 4 iω t}, z (t) = 0.

Therefore

∣ \dot{x} ∣ = ∣ \dot{ξ} ∣ = \frac{8∣ ω ∣}{3} ∣ sin (3 ω t) ∣.

The maximum speed and, for

ω > 0

, its times are

v_{m a x} = \frac{8 ω}{3}, t = \frac{( 2 k + 1 ) π}{6 ω} (k = 0, 1, 2, \dots) .

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