Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / Section B / B8 / c

To avoid contact with the ceiling, require $Y(\omega )<h$. Writing $X=r^2{\omega }^2$ giveswhose roots are $g(h\pm \sqrt{h^2-r^2})$. Because $h>r$, the lower root is less than $gr$, while the assumed branch has $X>gr$. The lower inequality is therefore automatic on this branch, leavingEquality is the first grazing contact. This branch check is essential: retaining both algebraic roots as independent lower and upper speed limits would introduce a spurious lower limit on avoiding the ceiling.