Challenging +1.2 This is a standard two-part circular motion problem requiring application of Newton's second law in polar form to find the separation condition (straightforward energy-force method), followed by projectile motion with restitution. The first part uses the standard template (energy equation + normal reaction = 0), and the second part is routine kinematics. While it requires multiple techniques and careful algebra, it follows well-established patterns for Further Maths mechanics questions without requiring novel insight.
A smooth sphere, with centre \(O\) and radius \(a\), is fixed on a smooth horizontal plane \(\Pi\). A particle \(P\) of mass \(m\) is projected horizontally from the highest point of the sphere with speed \(\sqrt { } \left( \frac { 2 } { 5 } g a \right)\). While \(P\) remains in contact with the sphere, the angle between \(O P\) and the upward vertical is denoted by \(\theta\). Show that \(P\) loses contact with the sphere when \(\cos \theta = \frac { 4 } { 5 }\).
Subsequently the particle collides with the plane \(\Pi\). The coefficient of restitution between \(P\) and \(\Pi\) is \(\frac { 5 } { 9 }\). Find the vertical height of \(P\) above \(\Pi\) when the vertical component of the velocity of \(P\) first becomes zero.
A smooth sphere, with centre $O$ and radius $a$, is fixed on a smooth horizontal plane $\Pi$. A particle $P$ of mass $m$ is projected horizontally from the highest point of the sphere with speed $\sqrt { } \left( \frac { 2 } { 5 } g a \right)$. While $P$ remains in contact with the sphere, the angle between $O P$ and the upward vertical is denoted by $\theta$. Show that $P$ loses contact with the sphere when $\cos \theta = \frac { 4 } { 5 }$.
Subsequently the particle collides with the plane $\Pi$. The coefficient of restitution between $P$ and $\Pi$ is $\frac { 5 } { 9 }$. Find the vertical height of $P$ above $\Pi$ when the vertical component of the velocity of $P$ first becomes zero.
\hfill \mbox{\textit{CAIE FP2 2013 Q11 EITHER}}