Define power as the rate of doing work or transferring energy, and apply $P = W / t = Fv$ to mechanical systems, including efficiency calculations

What this dot point is asking

QCAA wants you to use power as the rate quantity that links work and time, and to use efficiency as the ratio of useful energy output to total energy input. Both ideas appear in IA1 stimulus and EA Paper 1, often as small calculations attached to a larger motion or energy problem.

Power

Power is the rate at which work is done (or energy is transferred):

SI unit: watt (W = J s$^{-1}$). For a force pushing an object at constant velocity:

where $F$ is the component of force in the direction of motion. This form is useful when the force is constant and the speed is steady (cruising car, elevator at constant speed, conveyor belt).

Constant-power problems

When a vehicle delivers constant power, the driving force decreases as speed increases:

This is why a car has plenty of acceleration at low speeds but accelerates only slightly near top speed: the engine power is the same but $v$ is large.

At top speed (steady velocity), driving force equals total resistive force, so $P = F_{\text{resistive}} v_{\max}$.

Efficiency

Efficiency is the dimensionless ratio of useful output to total input:

Expressed as a fraction or a percentage. Always less than $1$ in real machines; the rest is energy lost as heat, sound, friction, or electrical resistance.

Connection to the rest of Unit 2

Power links to the work and mechanical energy dot point ($P = W/t$ converts a kinematic work calculation into a rate) and to motion graphs (constant-power motion gives a non-linear $v$-$t$ curve because acceleration depends on $v$). In Unit 3, the same power-efficiency framework reappears for transformers and AC transmission.

Try it: For a horizontal-motion power calculation, use the SUVAT calculator to get the steady velocity, then apply $P = Fv$ by hand.

Examples in context

Example 1. A Bundaberg sugar mill's bagasse-fired boiler delivers $40 \text{ MW}$ thermal at $80$ per cent boiler efficiency. The downstream turbine ($35$ per cent thermal-to-mechanical) produces $40 \times 0.80 \times 0.35 = 11.2 \text{ MW}$ shaft power, of which the generator converts $95$ per cent to electrical: $10.6 \text{ MW}$ delivered. Overall plant efficiency is $26.6$ per cent. The QCAA Unit 2 dot point's efficiency chain is exactly this product of stage efficiencies, a standard EA Paper 1 question.

Example 2. A Cairns light-rail induction motor produces $200 \text{ kW}$ traction power at $14 \text{ m s}^{-1}$, so it exerts $F = P/v = 14300 \text{ N}$ on the rail. Electrical input is $230 \text{ kW}$ from the $750 \text{ V}$ DC bus, giving motor efficiency $87$ per cent. Engineers monitor the $13$ per cent loss (mostly $I^2 R$ in windings, magnetic core loss, mechanical friction) to schedule maintenance; departures from baseline efficiency are a QCAA-style stimulus for IA1 data analysis.

Try this

Q1. Define power and state two equivalent algebraic forms. [2 marks]

• Cue. $P = W/t$; also $P = Fv$ at constant velocity.

Q2. A motor lifts a $200 \text{ kg}$ load $15 \text{ m}$ in $20 \text{ s}$. Calculate the useful output power. If input power is $1.8 \text{ kW}$, calculate the efficiency. [3 marks]

• Cue. $W = mgh = 29400 \text{ J}$; $P_{out} = 1470 \text{ W}$; efficiency $= 0.817 = 81.7$ per cent.

Q3. A Cairns tram induction motor delivers $200 \text{ kW}$ at $14 \text{ m s}^{-1}$. (a) Calculate the traction force. (b) Given $230 \text{ kW}$ electrical input, determine the efficiency and the power lost. (c) Identify two physical loss mechanisms with reasoning. [2+3+3 marks; ISMG: Knowledge and conceptual understanding, Evaluation]

• Cue. (a) $14300 \text{ N}$; (b) $87$ per cent, $30 \text{ kW}$ lost; (c) $I^2 R$ winding loss, magnetic core hysteresis-eddy losses.