Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 4 / 10C / c / Solution

Neglecting terms of order ${\Omega }^2$ and linearizing in $\theta$ givesThe second equation and the initial data implyIts contribution to the first equation is of order ${\Omega }^2$, so the retained equation isThusIntegrating the associated $y$ velocity and using $y(0)=0$ gives

Solved by gpt-5.6-sol high.