A Level Further Mathematics 9231 Paper 31, Oct/Nov 2025

Two uniform smooth spheres $A$ and $B$ of equal radii have masses $4m$ and $m$ respectively. Sphere $B$ is at rest on a smooth horizontal surface. Sphere $A$ is moving on the surface with speed $u$ and collides directly with sphere $B$. After the collision, the momentum of $A$ is three times the momentum of $B$. Find the value of the coefficient of restitution $e$.

A particle $P$ of mass $m$ is moving in a horizontal circle with angular speed ${\omega }_1$ on the smooth inner surface of a hemispherical shell of radius $r$. The angle between the upward vertical and the normal reaction of the surface on $P$ is ${\theta }_1$, where $\tan {\theta }_1=\frac{3}{4}$. When the angular speed is increased to ${\omega }_2$, the angle between the upward vertical and the normal reaction of the surface on $P$ becomes ${\theta }_2$, where $\tan {\theta }_2=\frac{4}{3}$. Find the ratio $\frac{{\omega }_1}{{\omega }_2}$.

A particle $P$ is moving in a straight horizontal line. At time $t\mathrm{s}$, the displacement of $P$ from a fixed point $O$ on the line is $x\mathrm{m}$ and the velocity of $P$ is $v{\mathrm{m}\mathrm{s}}^{-1}$. The acceleration of $P$ is $\frac{1}{2}(v^2+4){\mathrm{m}\mathrm{s}}^{-2}$ in the direction $PO$. Initially $P$ is at $O$ and is moving with velocity $2{\mathrm{m}\mathrm{s}}^{-1}$.

One end of a light elastic string of natural length $a$ and modulus of elasticity $5mg$ is attached to a fixed point $O$. Two particles, $P$ and $Q$, of masses $m$ and $4m$ respectively are attached to the other end of the string and they hang vertically in equilibrium. Particle $Q$ is then detached from the string, hence releasing particle $P$ from rest. Find, in terms of $a$, the length of the string when the speed of particle $P$ is first equal to $\sqrt{\frac{7}{5}ag}$.

A uniform lamina $OABCD$ consists of a rectangle $OACD$ and a triangle $ABC$. The length of $OA$ is $ka$, the length of $OD$ is $2a$, the height of triangle $ABC$ is $h$ and angle $CAB$ is ${45}^{\circ }$ (see diagram). Relative to axes through $O$, parallel and perpendicular to $OA$ as shown, the centre of mass of triangle $ABC$ is $(\bar{x},\bar{y})$.

A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. Initially $P$ is held with the string taut and making an angle of ${60}^{\circ }$ with the upward vertical through $O$. The particle $P$ is projected perpendicular to the string in a downwards direction with speed $\sqrt{17ag}$. It then starts to move along a circular path in a vertical plane with centre $O$ (see diagram). At the lowest point of its path, vertically below $O$, the particle $P$ collides with a stationary particle $Q$.

A particle $P$ is projected from a point $O$ on a horizontal plane and moves freely under gravity. The initial velocity of $P$ is $25{\mathrm{m}\mathrm{s}}^{-1}$ at an angle $\theta$ above the horizontal, where $\tan \theta =\frac{4}{3}$. At point $A$, the direction of motion of $P$ makes an angle of ${45}^{\circ }$ with the downward vertical through $A$.