A Level Further Mathematics 9231 Paper 31, May/June 2023

One end of a light elastic string, of natural length $a$ and modulus of elasticity $3mg$, is attached to a fixed point $O$. The other end of the string is attached to a particle $P$ of mass $m$. The string hangs with $P$ vertically below $O$. The particle $P$ is pulled vertically downwards so that the extension of the string is $2a$. The particle $P$ is then released from rest.

A particle $P$ of mass $m$ is moving with speed $u$ on a fixed smooth horizontal surface. It collides at an angle $\alpha$ with a fixed smooth vertical barrier. After the collision, $P$ moves at an angle $\theta$ with the barrier, where $\tan \theta =\frac{1}{2}$ (see diagram). The coefficient of restitution between $P$ and the barrier is $e$. The particle $P$ loses $20\mathrm{\%}$ of its kinetic energy as a result of the collision. Find the value of $e$.

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 $O$. The particle $P$ is held at the point $A$, where $OA$ makes an angle $\theta$ with the downward vertical through $O$, and with the string taut. The particle $P$ is projected perpendicular to $OA$ in an upwards direction with speed $u$. It then starts to move along a circular path in a vertical plane. The string goes slack when $P$ is at $B$, where angle $AOB$ is ${90}^{\circ }$ and the speed of $P$ is $\sqrt{\frac{4}{5}ag}$.

An object is formed from a solid hemisphere, of radius $2a$, and a solid cylinder, of radius $a$ and height $d$. The hemisphere and the cylinder are made of the same material. The cylinder is attached to the plane face of the hemisphere. The line $OC$ forms a diameter of the base of the cylinder, where $C$ is the centre of the plane face of the hemisphere and $O$ is common to both circumferences (see diagram). Relative to axes through $O$, parallel and perpendicular to $OC$ as shown, the centre of mass of the object is $(\bar{x},\bar{y})$.

A light elastic string of natural length $a$ and modulus of elasticity $\lambda mg$ has one end attached to a fixed point $O$ on a smooth horizontal surface. When a particle of mass $m$ is attached to the free end of the string, it moves with speed $v$ in a horizontal circle with centre $O$ and radius $x$. When, instead, a particle of mass $2m$ is attached to the free end of the string, this particle moves with speed $\frac{1}{2}v$ in a horizontal circle with centre $O$ and radius $\frac{3}{4}x$.

A particle $P$ moving in a straight line has displacement $x\mathrm{m}$ from a fixed point $O$ on the line and velocity $v{\mathrm{m}\mathrm{s}}^{-1}$ at time $t\mathrm{s}$. The acceleration of $P$, in ${\mathrm{m}\mathrm{s}}^{-2}$, is given by $6v\sqrt{v+9}$. When $t=0$, $x=2$ and $v=72$.

At time $t\mathrm{s}$, a particle $P$ is projected with speed $40{\mathrm{m}\mathrm{s}}^{-1}$ at an angle $\theta$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. The greatest height achieved by $P$ during its flight is $H\mathrm{m}$ and the corresponding time is $T\mathrm{s}$.