Energy, Work and Power

The aeroplane is flying horizontally. The aeroplane’s engines are producing a constant power of $5500\text{kW}$, and the aeroplane experiences a constant horizontal resistance force of $25\text{kN}$.

Find the speed of the aeroplane.

The aeroplane then ascends $300\text{m}$ in $50\text{s}$, while maintaining the same speed. The resistance force is no longer constant, and the work done against the resistance force in ascending the $300\text{m}$ is $270\,000\text{kJ}$. The mass of the aeroplane is $60\,000\text{kg}$.

Find the average power of the aeroplane’s engines.

The steady speed which the lorry can maintain with the engine working at power $P\text{ W}$ is $30\text{ m s}^{-1}$.

Find the value of $P$.

At an instant when the speed of the lorry is $16\text{ m s}^{-1}$, its engine is working at a power of $40\text{ kW}$.

Find the acceleration of the lorry at this instant.

When the lorry has reached a speed of $20\text{ m s}^{-1}$, it begins to ascend a section of road inclined at an angle $\alpha^\circ$ to the horizontal. The engine now works at a power of $120\text{ kW}$. There is no change in the lorry's speed as it ascends the hill. The constant resistance to motion remains $1600\text{ N}$.

Find the value of $\alpha$.

Find the value of $m$ and the value of $u$.

$P$ is now projected on a smooth horizontal surface with speed $v\text{ m s}^{-1}$ directly towards a particle $Q$ of mass $1.25\text{ kg}$ which is stationary. After $P$ and $Q$ collide, the velocity of $P$ is $w\text{ m s}^{-1}$ and the velocity of $Q$ is $2w\text{ m s}^{-1}$. The loss of kinetic energy in the collision is $25\text{ J}$.

Find the value of $v$ and the value of $w$.

The steady speed that the van could maintain when moving along a straight horizontal road is $50\text{ ms}^{-1}$.

Show that $k = 0.5$, and find the acceleration of the van when its speed is $25\text{ ms}^{-1}$ on this straight horizontal road.

The van begins to ascend a hill inclined at an angle $\theta^\circ$ to the horizontal. The van travels along a line of greatest slope of the hill. The speed of the van at the start of the hill is $20\text{ ms}^{-1}$, and its acceleration is $5a\text{ ms}^{-2}$. Later, on the same hill, the speed of the van is $30\text{ ms}^{-1}$, and its acceleration is $a\text{ ms}^{-2}$. The power of the van's engine remains at $62.5\text{ kW}$, and the resistance force remains at $0.5v^2\text{ N}$.

Find the value of $a$ and the value of $\theta$.

It is given that there is a constant resistance to motion. The engine of the car is working at $24\text{kW}$ while the car is travelling at a constant speed of $32\text{ms}^{-1}$. The power is now increased to $28\text{kW}$.

Find the acceleration of the car at the instant it is travelling at a speed of $36\text{ms}^{-1}$.

It is given instead that the resistance to the motion of the car is $(340 + 4v)\text{N}$ when the speed of the car is $v\text{ms}^{-1}$.

When the engine is working at $20\text{kW}$, the car is travelling at constant speed.

Find this constant speed.

Show that $c = 80$.

The motorcyclist now travels up a straight hill under maximum engine power. The hill makes an angle of $\sin^{-1}\left(\frac{3}{20}\right)$ with the horizontal.

Find the steady speed at which the motorcyclist travels up the hill.