Revise Work, energy and power for IB Physics (A.3)

In physics, work has a precise meaning: work is done when a force causes a displacement, and it is the mechanism by which energy is transferred to or from an object. Crucially, only the component of the force acting ALONG the displacement does work. A force applied to an object that does not move does no work; a force perpendicular to the motion — such as the normal force on a book sliding across a table — does no work either.

$W = Fs\cos\theta$

The formula $W = Fs\cos\theta$ only applies to a constant force. When the force VARIES with displacement — as a spring's force does — the work done is instead the AREA under the force–displacement ($F$–$s$) graph. This graphical view is completely general: for a constant force the area is simply a rectangle of height $F$ and width $s$, recovering $W = Fs$, while for a spring obeying $F = kx$ the area is a triangle, which is where the elastic energy formula comes from.

For mechanics you need three stores of energy. Kinetic energy ($E_k$) is the energy of motion — it grows with the square of the speed, so a car at 40 m s⁻¹ carries four times the kinetic energy it has at 20 m s⁻¹. Gravitational potential energy ($E_p$) is the energy of position in a gravitational field; lifting a mass raises it, and only the change in VERTICAL height matters. Elastic potential energy ($E_e$) is the energy stored in a stretched or compressed spring — it is the area under the spring's force–extension graph.

$E_k = \tfrac{1}{2}mv^2 \qquad \Delta E_p = mg\Delta h \qquad E_e = \tfrac{1}{2}kx^2$

The principle of conservation of energy states that in an isolated system the total energy is constant — energy is transferred between stores but is never created or destroyed. This is the single most powerful idea in mechanics, because it lets you jump directly from a starting state to a finishing state without tracking the forces at every instant. For a mass falling or sliding down a frictionless slope, gravitational potential energy is transferred entirely to kinetic energy: $mg\Delta h = \tfrac{1}{2}mv^2$. Notice the mass cancels, so the speed at the bottom does not depend on how heavy the object is.

When friction or air resistance is present, energy is not lost — it is transferred to thermal energy in the surfaces and surroundings. The bookkeeping simply gains one more term: the energy at the start equals the useful kinetic energy at the end PLUS the energy transferred to thermal energy. This is why a real object always arrives at the bottom of a rough slope slower than the frictionless prediction: some of its potential energy has been siphoned off as heat.

$mg\Delta h = \tfrac{1}{2}mv^2 + E_{\text{thermal}}$

Two machines can transfer the same energy, but the one that does it faster is more powerful. Power is the rate of doing work, or equivalently the rate of transferring energy, measured in watts (W), where one watt is one joule per second. For an object moving at a steady velocity $v$ under a constant force $F$, a compact form is available: since $W = Fs$ and $v = s/t$, we get $P = W/t = Fs/t = Fv$. This is ideal for steady-state motion such as a vehicle cruising against drag.

$P = \frac{W}{t} = \frac{\Delta E}{t} = Fv$

No real machine transfers all of its input energy into the store you want — some always ends up as unwanted thermal energy or sound. Efficiency measures how much of the input is usefully transferred: it is the ratio of useful output to total input, and because some energy is always wasted, it is always less than 1, or below 100%. You can compute it from energies or from powers, since power is just energy per unit time — as long as you compare like with like.

$\eta = \frac{\text{useful energy output}}{\text{total energy input}} = \frac{\text{useful power output}}{\text{total power input}}$

This is the showcase for a calculation topic. In Paper 2 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an answer mark (A) — and error-carried-forward (ECF) means a wrong number early on does not cost you every mark that follows, PROVIDED your method is written down. Notice how the marks here split between the numerical calculation and the physical REASONING about air resistance: the reasoning marks are earned by explanation, not arithmetic. Study how each mark is earned by a specific line.

Conservation of energy is not confined to this topic — it is the spine of the whole course. Momentum and collisions revisit kinetic energy to distinguish elastic from inelastic; simple harmonic motion swaps energy continually between elastic and kinetic stores; thermal physics tracks the very thermal energy that friction produces here; and in the study of fields, potential energy generalises from $mg\Delta h$ to gravitational and electric potentials. Master the habit of accounting — write every store at the start, every store at the end, set them equal — and problem after problem across the syllabus becomes a variation on a method you already own.