Particle $X$, of mass $2 \mathrm{kg}$, and particle $Y$, of mass $m \mathrm{kg}$, are attached to the ends of a light inextensible string of length $4.8 \mathrm{m}$. The string passes over a fixed small smooth pulley and hangs vertically either side of the pulley. Particle $X$ is held at ground level, $3 \mathrm{m}$ below the pulley. Particle $X$ is released and rises while particle $Y$ descends to the ground.

Find an expression, in terms of $m$, for the tension in the string while both particles are moving.

Particle $X$, of mass $2 \mathrm{kg}$, and particle $Y$, of mass $m \mathrm{kg}$, are attached to the ends of a light inextensible string of length $4.8 \mathrm{m}$. The string passes over a fixed small smooth pulley and hangs vertically either side of the pulley. Particle $X$ is held at ground level, $3 \mathrm{m}$ below the pulley. Particle $X$ is released and rises while particle $Y$ descends to the ground.

Use the work-energy principle to find how close particle $X$ gets to the pulley in the subsequent motion.

A car of mass $1000 \mathrm{kg}$ travels in a straight line up a slope inclined at angle $\alpha$ to the horizontal, where $\sin \alpha =0.05$. The non-gravitational resistances are $200 \mathrm{N}$ throughout the motion.

When the power produced by the engine is $50 \mathrm{kW}$, the car is accelerating at $1.2 \mathrm{m}{ \mathrm{s}}^{-2}$. Find the speed of the car at this instant.

A car of mass $1000 \mathrm{kg}$ travels in a straight line up a slope inclined at angle $\alpha$ to the horizontal, where $\sin \alpha =0.05$. The non-gravitational resistances are $200 \mathrm{N}$ throughout the motion.

When the power produced by the engine is $50 \mathrm{kW}$, the car is accelerating at $1.2 \mathrm{m}{ \mathrm{s}}^{-2}$.  What would happen to the speed if the mass of the car increased?

A car of mass $1000 \mathrm{kg}$ travels in a straight line up a slope inclined at angle $\alpha$ to the horizontal, where $\sin \alpha =0.05$. The non-gravitational resistances are $200 \mathrm{N}$ throughout the motion.

What would happen to the speed if the power produced by the engine decreased?

A car of mass $1200 \mathrm{kg}$ is driven along a straight horizontal road against a resistance of $5000 \mathrm{N}$. The engine has a maximum power output of $100 \mathrm{kW}$.

Find the maximum speed the car can reach.

A car of mass $1200 \mathrm{kg}$ is driven along a straight horizontal road against a resistance of $5000 \mathrm{N}$. The engine has a maximum power output of $100 \mathrm{kW}$.

Find the power being used when the car is travelling at a speed of $15{ \mathrm{ms}}^{-1}$ and accelerating at $-2{ \mathrm{ms}}^{-2}$.