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Past exam of the physics B course of the University of Cambridge / 2021 / Paper 1 / Section B / B6 / e / Solution 2026-10-18

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The original circular Kepler orbit has

v_{0}^{2} = GM / r_{0}

. For the transfer ellipse, set

r_{p} = r_{0}

r_{a} = 3 r_{0}

in part (c):

J_{t}^{2} = \frac{3}{2} GM m^{2} r_{0}, v_{p} = \frac{J _{t}}{m r _{0}} = \frac{3}{2} v_{0} .

(1)

The first required speed increase is

Δ v_{1} = (3/2 - 1) v_{0}

. Under the question's negligible-mass-change fuel model, fuel use is proportional to the magnitude of the impulse. Since the full fuel supply corresponds to

m v_{0}

f_{1} = \frac{m Δ v _{1}}{m v _{0}} = \frac{3}{2} - 1 = 0.2247 ≃ 22.5%.

(2)

This is the stated “about a quarter” of the original fuel supply. It is an instance of the Hohmann transfer impulse fractions; no variable-mass rocket model is required under the stipulated assumptions.

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