 Solution

ID: past-exam-of-the-short-options-paper-shared-by-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2019/s7/2/e/solution

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Past exam of the short options paper shared by Parts A and B of the physics course of the University of Oxford / 2019 / S7 / 2 / e / Solution by

Codex  0 2026-10-25

Let

m

be the ice mass for which the stated drag coefficients are defined,

K = k_{1} + k_{2}

and

f = 2Ω sin ℓ

. With stationary water, the steady horizontal ice equations are

0 = m f v_{y} + k_{1} v_{a, x} - K v_{x}, 0 = - m f v_{x} + k_{1} v_{a, y} - K v_{y} .

The linear system gives

v_{y} = \frac{k _{1}}{K ^{2} + ( m f ) ^{2}} (K v_{a, y} - m f v_{a, x}) .

For fixed wind speed

V = 10 m s^{- 1}

, the Cauchy-Schwarz inequality minimizes this component when

v_{a} = \frac{V}{K ^{2} + ( m f ) ^{2}} (m f, - K), v_{y, m i n} = - \frac{k _{1} V}{K ^{2} + ( m f ) ^{2}} .

Thus the wind must blow towards a direction east of south by

tan^{- 1} (m f / K)

in the northern hemisphere, west of south in the southern hemisphere, and due south at the equator. This compensates the Coriolis force; at the optimum

v_{x} = 0

and the ice moves due south. If the drag coefficients are per unit mass, replace

K / m

by the corresponding total drag rate. “Blows towards” avoids the opposite convention used when naming winds by their origin.

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