A Level Further Mathematics 9231 Paper 33, Oct/Nov 2023

Two uniform smooth spheres $A$ and $B$ of equal radii have masses $m$ and $2m$ respectively. The two spheres are moving with equal speeds $u$ on a smooth horizontal surface when they collide. Immediately before the collision, $A$'s direction of motion makes an angle of ${60}^{\circ }$ with the line of centres, and $B$'s direction of motion makes an angle $\theta$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $e$. After the collision, the component of the velocity of $A$ along the line of centres is $v$ and $B$ moves perpendicular to the line of centres. Sphere $A$ now has twice as much kinetic energy as sphere $B$.

A ball of mass $2\text{ kg}$ is projected vertically downwards with speed $5{\text{ ms}}^{-1}$ through a liquid. At time $t\text{ s}$ after projection, the velocity of the ball is $v{\text{ ms}}^{-1}$ and its displacement from its starting point is $x\text{ m}$. The forces acting on the ball are its weight and a resistive force of magnitude $0.2v^2\text{ N}$.

A uniform square lamina of side $2a$ and weight $W$ is suspended from a light inextensible string attached to the midpoint $E$ of the side $AB$. The other end of the string is attached to a fixed point $P$ on a rough vertical wall. The vertex $B$ of the lamina is in contact with the wall. The string $EP$ is perpendicular to the side $AB$ and makes an angle $\theta$ with the wall (see diagram). The string and the lamina are in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the lamina is $\frac{1}{2}$. Given that the vertex $B$ is about to slip up the wall, find the value of $\tan \theta$.

A light elastic string has natural length $8a$ and modulus of elasticity $5mg$. A particle $P$ of mass $m$ is attached to the midpoint of the string. The ends of the string are attached to points $A$ and $B$ which are a distance $12a$ apart on a smooth horizontal table. The particle $P$ is held on the table so that $AP=BP=L$ (see diagram). The particle $P$ is released from rest. When $P$ is at the midpoint of $AB$ it has speed $\sqrt{80ag}$.

A particle $P$ is projected with speed $u{\text{ ms}}^{-1}$ at an angle $\theta$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. During its flight $P$ passes through the point which is a horizontal distance $3a$ from $O$ and a vertical distance $\frac{3}{8}a$ above the horizontal plane. It is given that $\tan \theta =\frac{1}{3}$.

A particle $P$ of mass $m$ is attached to one end of a light inextensible rod of length $3a$. An identical particle $Q$ is attached to the other end of the rod. The rod is smoothly pivoted at a point $O$ on the rod, where $OQ=x$. The system, of rod and particles, rotates about $O$ in a vertical plane. At an instant when the rod is vertical, with $P$ above $Q$, the particle $P$ is moving horizontally with speed $u$. When the rod has turned through an angle of ${60}^{\circ }$ from the vertical, the speed of $P$ is $2\sqrt{ag}$, and the tensions in the two parts of the rod, $OP$ and $OQ$, have equal magnitudes.