A Level Further Mathematics 9231 Paper 33, May/June 2024

Two smooth uniform spheres $A$ and $B$ of equal radii have masses $m$ and $2m$ respectively. The two spheres are moving on a smooth horizontal surface when they collide with speeds $u$ and $\frac{1}{2}u$ respectively. Immediately before the collision, $A$'s direction of motion is along the line of centres, and $B$'s direction of motion makes an angle $\theta$ with the line of centres (see diagram). As a result of the collision, the direction of motion of $A$ is reversed and its speed is reduced to $\frac{1}{4}u$. The direction of motion of $B$ again makes an angle $\theta$ with the line of centres, but on the opposite side of the line of centres. The speed of $B$ is unchanged. Find the value of the coefficient of restitution between the spheres.

A particle $P$ of mass $m$ is attached to one end of a light elastic string of natural length $a$ and modulus of elasticity $2mg$. A particle $Q$ of mass $km$ is attached to the other end of the string. Particle $P$ lies on a smooth horizontal table. The string has part of its length in contact with the table and then passes through a small smooth hole $H$ in the table. Particle $P$ moves in a horizontal circle on the surface of the table with constant speed $\sqrt{\frac{1}{2}ga}$. Particle $Q$ hangs in equilibrium vertically below the hole with $HQ=\frac{1}{4}a$.

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 $O$. When the particle is hanging vertically below $O$, it is projected horizontally with speed $u$ so that it begins to move along a circular path. When $P$ is at the lowest point of its motion, the tension in the string is $T$. When $OP$ makes an angle $\theta$ with the upward vertical, the tension in the string is $S$.

A light spring of natural length $a$ and modulus of elasticity $kmg$ is attached to a fixed point $O$ on a smooth plane inclined to the horizontal at an angle $\theta$, where $\sin \theta =\frac{3}{4}$. A particle of mass $m$ is attached to the lower end of the spring and is held at the point $A$ on the plane, where $OA=2a$ and $OA$ is along a line of greatest slope of the plane (see diagram). The particle is released from rest and is moving with speed $V$ when it passes through the point $B$ on the plane, where $OB=\frac{3}{2}a$. The speed of the particle is $\frac{1}{2}V$ when it passes through the point $C$ on the plane, where $OC=\frac{3}{4}a$. Find the value of $k$.

A uniform lamina is in the form of a triangle $OBC$, with $OC=18a$, $OB=24a$ and angle $COB={90}^{\circ }$. The point $A$ on $OB$ is such that $OA=x$ (see diagram). The triangle $OAC$ is removed from the lamina.

A particle $P$ is projected with speed $u{\text{ ms}}^{-1}$ at an angle $\theta$ above the horizontal from a point $O$ and moves freely under gravity. After 5 seconds the speed of $P$ is $\frac{3}{4}u$.

A parachutist of mass $m\text{ kg}$ opens his parachute when he is moving vertically downwards with a speed of $50{\text{ ms}}^{-1}$. At time $t\text{ s}$ after opening his parachute, he has fallen a distance $x\text{ m}$ from the point where he opened his parachute, and his speed is $v{\text{ ms}}^{-1}$. The forces acting on him are his weight and a resistive force of magnitude $mv\text{ N}$.