Revise Energy, work and power for Cambridge 9709 (4.5)

In physics, 'work' has a very specific meaning. Work is done by a force when it causes an object to move a certain distance. Crucially, only the component of the force that acts in the direction of motion contributes to the work done. If you push a box horizontally, your entire force does work. If you pull it with a rope at an angle, only the horizontal component of your tension does work.

$W = Fd \cos \theta$ Where $W$ is the work done in Joules (J), $F$ is the magnitude of the constant force in Newtons (N), $d$ is the magnitude of the displacement in metres (m), and $\theta$ is the angle between the force vector and the displacement vector.

Energy is the capacity to do work. In A-Level Mechanics, we are primarily concerned with two types of mechanical energy: Kinetic Energy and Gravitational Potential Energy.

Kinetic Energy (KE): The energy an object possesses due to its motion. $KE = \frac{1}{2}mv^2$ Gravitational Potential Energy (GPE): The energy an object possesses due to its position in a gravitational field, relative to a reference level. $GPE = mgh$

For GPE, $h$ is the vertical height above a chosen 'zero' level. The choice of this zero level is arbitrary, but you must be consistent throughout a problem. Often, the lowest point in the object's motion is the most convenient choice for $h=0$.

These principles connect work and energy. The most general form is the Work-Energy Principle: the total work done by all forces on an object equals the change in its kinetic energy. A special, very useful case is the Principle of Conservation of Mechanical Energy. This applies when there are no non-conservative forces doing work (like friction or air resistance) and no external driving forces. In such cases, energy simply converts between kinetic and potential forms, but the total amount (KE + GPE) remains constant.

Power is the rate at which work is done. A powerful engine can do a lot of work in a short amount of time. The SI unit of power is the Watt (W), where 1 Watt is equal to 1 Joule per second (1 W = 1 J/s). For vehicles, power is often quoted in kilowatts (kW) or horsepower (hp).

Average Power: $P_{avg} = \frac{\text{Work Done}}{\text{Time Taken}} = \frac{W}{t}$ Instantaneous Power: $P = Fv$ Where $F$ is the driving force and $v$ is the instantaneous velocity. This second formula is extremely useful for problems involving vehicles with a constant power output.