Challenging +1.8 This is a challenging Further Maths mechanics problem requiring multiple sophisticated steps: deriving the loss-of-contact condition using circular motion dynamics and energy conservation, then analyzing post-collision motion with coefficient of restitution. The problem demands strong conceptual understanding of normal reaction forces, energy methods, and projectile motion, going well beyond standard A-level mechanics questions.
11 Answer only one of the following two alternatives.
EITHER
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.
OR
A factory produces bottles of spring water. The manager decides to assess the performance of the two machines that are used to fill the bottles with water. He selects a random sample of 60 bottles filled by the first machine \(X\) and a random sample of 80 bottles filled by the second machine \(Y\). The volumes of water, \(x\) and \(y\), measured in appropriate units, are summarised as follows.
$$\Sigma x = 58.2 \quad \Sigma x ^ { 2 } = 85.8 \quad \Sigma y = 97.6 \quad \Sigma y ^ { 2 } = 188.6$$
A test at the \(\alpha \%\) significance level shows that the mean volume of water in bottles filled by machine \(X\) is less than the mean volume of water in bottles filled by machine \(Y\). Find the set of possible values of \(\alpha\).
11 Answer only one of the following two alternatives.
\\EITHER\\
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.
\\OR\\
A factory produces bottles of spring water. The manager decides to assess the performance of the two machines that are used to fill the bottles with water. He selects a random sample of 60 bottles filled by the first machine $X$ and a random sample of 80 bottles filled by the second machine $Y$. The volumes of water, $x$ and $y$, measured in appropriate units, are summarised as follows.
$$\Sigma x = 58.2 \quad \Sigma x ^ { 2 } = 85.8 \quad \Sigma y = 97.6 \quad \Sigma y ^ { 2 } = 188.6$$
A test at the $\alpha \%$ significance level shows that the mean volume of water in bottles filled by machine $X$ is less than the mean volume of water in bottles filled by machine $Y$. Find the set of possible values of $\alpha$.
\hfill \mbox{\textit{CAIE FP2 2013 Q11}}