A Level Mathematics 9709 Paper 41, Oct/Nov 2023

A particle of mass $1.6\text{ kg}$ is projected with a speed of $20{\text{ m s}}^{-1}$ up a line of greatest slope of a smooth plane inclined at $\alpha$ to the horizontal, where $\tan \alpha =\frac{3}{4}$. Use an energy method to find the distance the particle moves up the plane before coming to instantaneous rest.

A particle of mass $2.4\text{ kg}$ is held in equilibrium by two light inextensible strings, one of which is attached to point $A$ and the other attached to point $B$. The strings make angles of ${35}^{\circ }$ and ${40}^{\circ }$ with the horizontal (see diagram). Find the tension in each of the two strings.

The diagram shows the velocity-time graph for the motion of a bus. The bus starts from rest and accelerates uniformly for $8\text{ seconds}$ until it reaches a speed of $12.6{\text{ m s}}^{-1}$. The bus maintains this speed for $40\text{ seconds}$. It then decelerates uniformly in two stages. Between $48$ and $62\text{ seconds}$ the bus decelerates at $a{\text{ m s}}^{-2}$ and between $62$ and $70\text{ seconds}$ it decelerates at $2a{\text{ m s}}^{-2}$ until coming to rest.

Two particles $P$ and $Q$, of masses $6\text{ kg}$ and $2\text{ kg}$ respectively, lie at rest $12.5\text{ m}$ apart on a rough horizontal plane. The coefficient of friction between each particle and the plane is $0.4$. Particle $P$ is projected towards $Q$ with speed $20{\text{ m s}}^{-1}$.

The diagram shows a particle $A$, of mass $1.2\text{ kg}$, which lies on a plane inclined at an angle of ${40}^{\circ }$ to the horizontal and a particle $B$, of mass $1.6\text{ kg}$, which lies on a plane inclined at an angle of ${50}^{\circ }$ to the horizontal. The particles are connected by a light inextensible string which passes over a small smooth pulley $P$ fixed at the top of the planes. The parts $AP$ and $BP$ of the string are taut and parallel to lines of greatest slope of the respective planes. The two planes are rough, with the same coefficient of friction, $\mu$, between the particles and the planes. Find the value of $\mu$ for which the system is in limiting equilibrium.

A car of mass $1300\text{ kg}$ is moving on a straight road.

A particle moves in a straight line starting from a point $O$ before coming to instantaneous rest at a point $X$. At time $t\text{ s}$ after leaving $O$, the velocity $v{\text{ m s}}^{-1}$ of the particle is given by $$\begin{array}{rlr}{\displaystyle v}& {\displaystyle =7.2t^2}& {\displaystyle 0⩽t⩽2,}\\ {\displaystyle v}& {\displaystyle =30.6-0.9t}& {\displaystyle 2⩽t⩽8,}\\ {\displaystyle v}& {\displaystyle =\frac{1600}{t^2}+kt}& {\displaystyle 8⩽t,}\end{array}$$ where $k$ is a constant. It is given that there is no instantaneous change in velocity at $t=8$. Find the distance $OX$.