Revise Energy conservation for Cambridge 9702 (5.1)

At its core, the Principle of Conservation of Energy states that energy can neither be created nor destroyed. Instead, it continuously changes from one form to another. In a closed system, where no energy can enter or leave, the total amount of energy always stays constant.

Two crucial forms of energy we frequently encounter are Kinetic Energy (KE) and Gravitational Potential Energy (GPE). KE is the energy an object possesses due to its motion, while GPE is the energy stored due to its position in a gravitational field.

$KE = \frac{1}{2}mv^2$

Kinetic energy depends on an object's mass ($m$) and the square of its velocity ($v$). The faster or more massive an object, the more kinetic energy it has.

$GPE = mgh \text{ or } \Delta GPE = mg\Delta h$

Gravitational potential energy depends on mass ($m$), gravitational acceleration ($g$), and vertical height ($h$). You can choose any convenient reference point for 'h', as only the change in height matters for $\Delta GPE$.

When energy is transferred, we call it work done. Work is done when a force causes an object to move a certain distance. Think of pushing a box across a floor – you're doing work on it. If the force and displacement are in the same direction, the calculation is simple:

$W = Fd$

What if the force is not in the same direction as the displacement? For example, pulling a suitcase with a handle at an angle. In this case, only the component of the force that acts in the direction of motion does work. We use the formula:

$W = Fd \cos \theta$

Here, $\theta$ is the angle between the force vector and the displacement vector. If the force is perpendicular to the displacement ($\theta = 90°$), then $\cos(90°) = 0$, and no work is done. This is why the gravitational force does no work on an object moving horizontally.

Power is how quickly this energy transfer or work happens. A powerful engine can do a lot of work in a short amount of time, indicating a high rate of energy transfer.

$P = \frac{\Delta E}{\Delta t} \text{ and } P = Fv$

The formula $P = Fv$ is particularly useful for objects moving at a constant velocity $v$ against a constant force $F$. It's derived from $P = W/t = (Fd)/t$ and knowing that $v = d/t$ for constant velocity.

Finally, efficiency tells us how effectively a system converts input energy into useful output energy. No real system is 100% efficient because some energy is always 'lost' to unwanted forms, typically heat or sound. For example, an electric motor lifting a mass gains useful GPE, but some input electrical energy is wasted as heat in the motor's coils and sound from its operation.

$Efficiency = \frac{\text{Useful Energy Output}}{\text{Total Energy Input}} = \frac{\text{Useful Power Output}}{\text{Total Power Input}}$