Work, energy and power

In physics, 'work' has a very specific meaning. Work is done whenever a force causes an object to move. It is the transfer of energy from one form to another. For work to be done, two conditions must be met: there must be a force, and the object must move in the direction of the force. The amount of work done is calculated as the product of the force and the distance moved in the direction of the force. If you push against a wall and it doesn't move, you have done no work on the wall, even though you might feel tired! The unit of work is the Joule (J). One Joule is the work done when a force of 1 Newton moves an object by 1 metre. If the force acts at an angle to the direction of motion, we only consider the component of the force that is in the direction of motion. This is calculated using `W = Fs cos(θ)`, where θ is the angle between the force and the displacement.

Gravitational Potential Energy (GPE) is the energy an object has stored because of its position in a gravitational field. When you lift an object, you do work against the force of gravity. This work done is converted into GPE and stored in the object. To lift an object of mass 'm' to a vertical height 'h', you must apply a force at least equal to its weight, which is 'mg' (where 'g' is the acceleration of free fall, approximately 9.81 m/s²). The work done is Force × distance, so `W = (mg) × h`. This work done is equal to the gain in GPE. Therefore, the change in gravitational potential energy is given by the formula `ΔEp = mgΔh`. Note that it is the change in vertical height that matters, not the path taken to get there.

Kinetic Energy (KE) is the energy an object possesses because of its motion. Anything that is moving has kinetic energy, from a moving car to a tiny atom. The amount of kinetic energy an object has depends on two things: its mass(m) and its speed (v). The formula for kinetic energy is `Ek = ½mv²`. This formula can be derived from the work-energy principle. The work done on an object by a net force causes a change in its kinetic energy. If a force 'F' acts on a mass 'm' over a distance 's', the work done is `W = Fs`. Using Newton's second law, `F = ma`, so `W = mas`. From the equations of motion, `v² = u² + 2as`. If the object starts from rest (u=0), then `v² = 2as`, so `as = v²/2`. Substituting this into the work equation gives `W = m(v²/2)`, which is `W = ½mv²`. This work done is equal to the kinetic energy gained. Notice the `v²` term: this means that doubling the speed of an object quadruples its kinetic energy, which is why high-speed collisions are so devastating.

This is one of the most fundamental principles in all of physics. The Principle of Conservation of Energy states that in an isolated system, energy cannot be created or destroyed; it can only be converted from one form to another. The total amount of energy in the system remains constant. For example, when a ball is dropped, its gravitational potential energy (GPE) is converted into kinetic energy (KE) as it falls and speeds up. If we ignore air resistance, the GPE lost is exactly equal to the KE gained. In a real-world scenario, some energy is always converted into less useful forms, like heat and sound, due to forces like friction and air resistance. The principle still holds true; the initial energy just gets distributed among more forms. So, a more complete equation is: `Initial Energy = Final Useful Energy + Energy dissipated (e.g. as heat)`.

While energy is always conserved, it is not always converted into the form we want. When we use a machine, like a light bulb or a car engine, we provide it with an energy input (e.g., electrical energy, chemical energy from fuel). The machine converts this into a useful energy output (e.g., light, kinetic energy) but also produces wasted energy, usually as heat and sound. Efficiency is a measure of how good a device is at converting the total energy input into useful energy output. It is calculated as a ratio, often expressed as a percentage. A perfectly efficient machine would have an efficiency of 100%, but this is impossible in practice due to the second law of thermodynamics. Improving efficiency is a major goal in engineering to save energy and reduce costs.

In physics, power is not about strength, but about how quickly energy is used or work is done. A powerful machine is one that can transfer a large amount of energy in a short amount of time. Power is defined as the rate of work done or the rate of energy transfer. The standard unit for power is the Watt (W), named after the Scottish engineer James Watt. One Watt is defined as a rate of energy transfer of one Joule per second (1 W = 1 J/s). So, a 100 W light bulb converts 100 Joules of electrical energy into light and heat every second. The main formula for power is `P = W/t` (Power = Work done / time taken) or `P = ΔE/t` (Power = Energy transferred / time taken).

There is a very useful relationship between power, force, and velocity, especially for vehicles like cars, trains, and planes. We can derive it from the definition of power. We know that `P = W/t` and `W = Fs`. Substituting the work formula into the power formula gives `P = (Fs)/t`. We can group the terms as `P = F × (s/t)`. Since displacement divided by time (`s/t`) is velocity (`v`), we arrive at the equation `P = Fv`. This equation tells us the power `P` required to provide a driving force `F` to an object moving at a constant velocity `v`. In this context, `F` is the driving force that is overcoming resistive forces like friction and air resistance. At a constant velocity, the driving force is equal to the total resistive forces.