A Level Mathematics 9709 Paper 42, Oct/Nov 2024

The velocity of a particle moving in a straight line at time $t$ seconds after leaving a fixed point $O$ is $v{\mathrm{m}\mathrm{s}}^{-1}$. The diagram shows a velocity-time graph which models the motion of the particle from $t=0$ to $t=T$. The graph consists of four straight line segments. The particle accelerates from rest to a speed of $V{\mathrm{m}\mathrm{s}}^{-1}$ over a period of $4\mathrm{s}$, and then decelerates at $\frac{5}{3}{\mathrm{m}\mathrm{s}}^{-2}$ to instantaneous rest over a period of $6\mathrm{s}$. The particle then travels back towards $O$, reaching a maximum speed of $3{\mathrm{m}\mathrm{s}}^{-1}$ before coming to rest at time $t=T$.

A block of mass $20\mathrm{k}\mathrm{g}$ is held at rest at the top of a plane inclined at ${30}^{\circ }$ to the horizontal. The block is projected with speed $5{\mathrm{m}\mathrm{s}}^{-1}$ down a line of greatest slope of the plane. There is a resistance force acting on the block. As the block moves $2\mathrm{m}$ down the plane from its point of projection, the work done against this resistance force is $50\mathrm{J}$. Find the speed of the block when it has moved $2\mathrm{m}$ down the plane.

A cyclist is riding along a straight horizontal road. The total mass of the cyclist and his bicycle is $90\mathrm{k}\mathrm{g}$. The power exerted by the cyclist is $250\mathrm{W}$. At an instant when the cyclist’s speed is $5{\mathrm{m}\mathrm{s}}^{-1}$, his acceleration is $0.1{\mathrm{m}\mathrm{s}}^{-2}$.

The diagram shows two particles, $A$ and $B$, of masses $0.2\mathrm{k}\mathrm{g}$ and $0.1\mathrm{k}\mathrm{g}$ respectively. The particles are suspended below a horizontal ceiling by two strings, $AP$ and $BQ$, attached to fixed points $P$ and $Q$ on the ceiling. The particles are connected by a horizontal string, $AB$. Angle $APQ={45}^{\circ }$ and $BQP={\theta }^{\circ }$. Each string is light and inextensible. The particles are in equilibrium.

Two particles, $P$ and $Q$, of masses $2m\mathrm{k}\mathrm{g}$ and $m\mathrm{k}\mathrm{g}$ respectively, are held at rest in the same vertical line. The heights of $P$ and $Q$ above horizontal ground are $1\mathrm{m}$ and $2\mathrm{m}$ respectively. $P$ is projected vertically upwards with speed $2{\mathrm{m}\mathrm{s}}^{-1}$. At the same instant, $Q$ is released from rest.

A particle, $P$, travels in a straight line, starting from a point $O$ with velocity $6{\mathrm{m}\mathrm{s}}^{-1}$. The acceleration of $P$ at time $t\mathrm{s}$ after leaving $O$ is $a{\mathrm{m}\mathrm{s}}^{-2}$, where $$\begin{array}{rl}{\displaystyle a}& {\displaystyle =-1.5t^{\frac{1}{2}}\text{for }0⩽t⩽1,}\\ {\displaystyle a}& {\displaystyle =1.5t^{\frac{1}{2}}-3t^{-\frac{1}{2}}\text{for }t>1.}\end{array}$$

Two particles, $A$ and $B$, of masses $0.2\mathrm{k}\mathrm{g}$ and $0.3\mathrm{k}\mathrm{g}$ respectively, are attached to the ends of a light inextensible string. The string passes over a small fixed smooth pulley which is attached to the bottom of a rough plane inclined at an angle $\theta$ to the horizontal where $\sin \theta =0.6$. Particle $A$ lies on the plane, and particle $B$ hangs vertically below the pulley, $0.25\mathrm{m}$ above horizontal ground. The string between $A$ and the pulley is parallel to a line of greatest slope of the plane (see diagram). The coefficient of friction between $A$ and the plane is $1.125$. Particle $A$ is released from rest.