Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 316 / 1 / iv

Δ v_{p c}

b

h = b Δ v_{p c} = a_{m} v_{c, t}, v_{c, t} = \frac{b Δ v _{p c}}{a _{m}} .

v_{c, r}^{2} + v_{c, t}^{2} = Δ v_{p c}^{2} + \frac{2 G M _{p}}{a _{m}} = Δ v_{p c}^{2} + 2 v_{m}^{2},

where

v_{m}^{2} = G M_{p} / a_{m}

. Thus

v_{c, r} = \pm [Δ v_{p c}^{2} + 2 v_{m}^{2} - (\frac{b Δ v _{p c}}{a _{m}})^{2}]^{1/2} .

The moon moves tangentially at

v_{m}

, so

Δ v_{m c}^{2} = v_{c, r}^{2} + (v_{c, t} - v_{m})^{2} = 3 v_{m}^{2} - 2 v_{m} Δ v_{p c} \frac{b}{a _{m}} + Δ v_{p c}^{2} .

Therefore the correctly dimensioned reading of the displayed result is

Δ v_{m c} = (3 v_{m}^{2} - 2 v_{m} Δ v_{p c} \frac{b}{a _{m}} + Δ v_{p c}^{2})^{1/2} .

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