Car $A$, of mass $1250 \mathrm{kg}$, is travelling along a straight horizontal road at speed $10 \mathrm{m}{ \mathrm{s}}^{-1}$. The engine works at a constant rate of $25 \mathrm{kW}$ and the resistance is a constant $500 \mathrm{N}$. After $5 \mathrm{s}$ the speed of the car has increased to $v \mathrm{m}{ \mathrm{s}}^{-1}$.

Use the work-energy principle to find the amount of energy that is dissipated and, hence, find the distance travelled in the $5 \mathrm{s}$.

Car $A$, of mass $1250 \mathrm{kg}$, is travelling along a straight horizontal road at speed $10 \mathrm{m}{ \mathrm{s}}^{-1}$. The engine works at a constant rate of $25 \mathrm{kW}$ and the resistance is a constant $500 \mathrm{N}$. After $5 \mathrm{s}$ the speed of the car has increased to $v \mathrm{m}{ \mathrm{s}}^{-1}$.

Find an expression for the acceleration at time $5 \mathrm{s}$ as a function of $v$ and show that the acceleration is not constant.

Car $A$, of mass $1250 \mathrm{kg}$, is travelling along a straight horizontal road at speed $10 \mathrm{m}{ \mathrm{s}}^{-1}$. The engine works at a constant rate of $25 \mathrm{kW}$ and the resistance is a constant $500 \mathrm{N}$. After $5 \mathrm{s}$ the speed of the car has increased to $v \mathrm{m}{ \mathrm{s}}^{-1}$.

Car B travels along the same road, starting with speed $10 m{s}^{-1}$ and accelerating at a constant rate for$5s$ . After$5s$  the two cars have the same speed and also have the same acceleration as one another.

Show that $v$ must satisfy the equation $v^2-8v=100$ and, hence, find the speed of the cars at the end of the $5 \mathrm{s}$.

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?