A Level Further Mathematics 9231 Paper 31, Oct/Nov 2024

A particle $P$ is projected with speed $u\mathrm{m}{\mathrm{s}}^{-1}$ at an angle ${\tan }^{-1}2$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. When $P$ has travelled a distance $56\mathrm{m}$ horizontally from $O$, it is at a vertical height $H\mathrm{m}$ above the plane. When $P$ has travelled a distance $84\mathrm{m}$ horizontally from $O$, it is at a vertical height $\frac{1}{2}H\mathrm{m}$ above the plane. Find, in either order, the value of $u$ and the value of $H$.

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 $A$ with the string taut. It is given that $OA$ makes an angle $\theta$ with the downward vertical through $O$, where $\tan \theta =\frac{3}{4}$. The particle $P$ is projected perpendicular to $OA$ in an upwards direction with speed $\sqrt{5ag}$, and it starts to move along a circular path in a vertical plane. When $P$ is at the point $B$, where angle $AOB$ is a right angle, the tension in the string is $T$. Find $T$ in terms of $m$ and $g$.

A particle $P$ of mass $m\mathrm{k}\mathrm{g}$ is attached to one end of a light elastic string of natural length $2\mathrm{m}$ and modulus of elasticity $2mg\mathrm{N}$. The other end of the string is attached to a fixed point $O$. The particle $P$ hangs in equilibrium vertically below $O$. The particle $P$ is pulled down vertically a distance $d\mathrm{m}$ below its equilibrium position and released from rest.

An object is formed by removing a cylinder of radius $\frac{2}{3}a$ and height $kh$ ($k<1$) from a uniform solid cylinder of radius $a$ and height $h$. The vertical axes of symmetry of the two cylinders coincide. The upper faces of the two cylinders are in the same plane as each other. The points $A$ and $B$ are the opposite ends of a diameter of the upper face of the object (see diagram).

A particle $P$ of mass $2\mathrm{k}\mathrm{g}$ moving on a horizontal 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 only horizontal force acting on $P$ is a variable force $F\mathrm{N}$ which can be expressed as a function of $t$. It is given that $$\frac{v}{x}=\frac{3-t}{1+t}$$ and when $t=0$, $x=5$.

A particle $P$ of mass $0.05\mathrm{k}\mathrm{g}$ is attached to one end of a light inextensible string of length $1\mathrm{m}$. The other end of the string is attached to a fixed point $O$. A particle $Q$ of mass $0.04\mathrm{k}\mathrm{g}$ is attached to one end of a second light inextensible string. The other end of this string is attached to $P$. The particle $P$ moves in a horizontal circle of radius $0.8\mathrm{m}$ with angular speed $\omega \mathrm{r}\mathrm{a}\mathrm{d}{\mathrm{s}}^{-1}$. The particle $Q$ moves in a horizontal circle of radius $1.4\mathrm{m}$ also with angular speed $\omega \mathrm{r}\mathrm{a}\mathrm{d}{\mathrm{s}}^{-1}$. The centres of the circles are vertically below $O$, and $O$, $P$ and $Q$ are always in the same vertical plane. The strings $OP$ and $PQ$ remain at constant angles $\alpha$ and $\beta$ respectively to the vertical (see diagram).

A particle $P$ is projected with speed $u$ at an angle ${\tan }^{-1}\left(\frac{4}{3}\right)$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. When $P$ is moving horizontally, it strikes a smooth inclined plane at the point $A$. This plane is inclined to the horizontal at an angle $\alpha$, and the line of greatest slope through $A$ lies in the vertical plane through $O$ and $A$. As a result of the impact, $P$ moves vertically upwards. The coefficient of restitution between $P$ and the inclined plane is $e$.