A Level Mathematics 9709 Paper 43, May/June 2023

Two particles $P$ and $Q$, of masses $0.1\text{ kg}$ and $0.4\text{ kg}$ respectively, are free to move on a smooth horizontal plane. Particle $P$ is projected with speed $4{\text{ m s}}^{-1}$ towards $Q$ which is stationary. After $P$ and $Q$ collide, the speeds of $P$ and $Q$ are equal. Find the two possible values of the speed of $P$ after the collision. [3]

A car of mass $1500\text{ kg}$ is towing a trailer of mass $m\text{ kg}$ along a straight horizontal road. The car and the trailer are connected by a tow-bar which is horizontal, light and rigid. There is a resistance force of $F\text{ N}$ on the car and a resistance force of $200\text{ N}$ on the trailer. The driving force of the car’s engine is $3200\text{ N}$, the acceleration of the car is $1.25{\text{ m s}}^{-2}$ and the tension in the tow-bar is $300\text{ N}$. Find the value of $m$ and the value of $F$. [4]

A smooth ring $R$ of mass $0.2\text{ kg}$ is threaded on a light string $ARB$. The ends of the string are attached to fixed points $A$ and $B$ with $A$ vertically above $B$. The string is taut and angle $ABR={90}^{\circ }$. The angle between the part $AR$ of the string and the vertical is ${60}^{\circ }$. The ring is held in equilibrium by a force of magnitude $X\text{ N}$, acting on the ring in a direction perpendicular to $AR$ (see diagram). Calculate the tension in the string and the value of $X$. [5]

A lorry of mass $15000\text{ kg}$ moves on a straight horizontal road in the direction from $A$ to $B$. It passes $A$ and $B$ with speeds $20{\text{ m s}}^{-1}$ and $25{\text{ m s}}^{-1}$ respectively. The power of the lorry’s engine is constant and there is a constant resistance to motion of magnitude $6000\text{ N}$. The acceleration of the lorry at $B$ is $0.5$ times the acceleration of the lorry at $A$.

A particle starts from rest from a point $O$ and moves in a straight line. The acceleration of the particle at time $t\text{ s}$ after leaving $O$ is $a{\text{ m s}}^{-2}$, where $a=k{t}^{\frac{1}{2}}$ for $0⩽t⩽9$ and where $k$ is a constant. The velocity of the particle at $t=9$ is $1.8{\text{ m s}}^{-1}$.

An elevator is pulled vertically upwards by a cable. The elevator accelerates at $0.4{\text{ m s}}^{-2}$ for $5\text{ s}$, then travels at constant speed for $25\text{ s}$. The elevator then decelerates at $0.2{\text{ m s}}^{-2}$ until it comes to rest.

The diagram shows the vertical cross-section $XYZ$ of a rough slide. The section $YZ$ is a straight line of length $2\text{ m}$ inclined at an angle of $\alpha$ to the horizontal, where $\sin \alpha =0.28$. The section $YZ$ is tangential to the curved section $XY$ at $Y$, and $X$ is $1.8\text{ m}$ above the level of $Y$. A child of mass $25\text{ kg}$ slides down the slide, starting from rest at $X$. The work done by the child against the resistance force in moving from $X$ to $Y$ is $50\text{ J}$.