Learning Objectives
3 objectivesBy the end of this note, you should be able to:
- Define power as work done per unit time and as energy transferred per unit time.
- Recall and use the equation P = W / t.
- Recall and use the equation P = ΔE / t.
Defining Power
Power is the rate of doing work, or equivalently, the rate of energy transfer.
Two machines may each do the same amount of work, yet one finishes faster. The faster machine has a greater power because it transfers the same energy in a shorter time. Power therefore measures how quickly energy is transferred from one store to another — not how much energy is transferred in total.
The SI unit of power is the watt (W). One watt equals one joule of work done (or energy transferred) per second, so 1 W = 1 J s⁻¹. A kilowatt (kW) equals 1000 W and a megawatt (MW) equals 1 000 000 W.
MisconceptionStudents sometimes confuse power with energy. A device that transfers a large amount of energy is not necessarily high-powered — it may simply have been running for a long time. Power depends on how fast the energy is transferred, not just the total amount. Exam cue: always check whether a question asks for energy (J) or power (W).
Calculating Power from Work Done
Key Equations
Power (from work done):
$$P=\frac{W}{t}$$
Variables:
- $P$ = power, in watts (W)
- $W$ = work done, in joules (J)
- $t$ = time taken, in seconds (s)
SI unit of calculated quantity: W (watt)
Rearrangements
Starting from $P=\frac{W}{t}$:
Rearranging for work done:
$$W=P\times t$$
Rearranging for time:
$$t=\frac{W}{P}$$
ProportionalityPower is directly proportional to work done when time is constant — doubling the work done doubles the power. Power is inversely proportional to time when work done is constant — halving the time doubles the power.
Reading the equation: The symbol $\Delta $ (Greek capital delta) means "change in." The fraction bar means "divided by." So $P=\frac{W}{t}$ reads: power equals work done divided by time.
Worked Example
A crane lifts a steel beam, doing 24 000 J of work in 1.5 minutes. Calculate the power of the crane.
Converting time to SI units
$$t=1.5\times 60$$
$$t=90\text{ s}$$
Finding the power
Equation used — Power (from work done)
$$P=\frac{W}{t}$$
Given
$$W=24 000\text{ J}$$
$$t=90\text{ s}$$
Substituting:
$$P=\frac{24 000}{90}$$
$$P = 266.6\ldots \text{ W}$$
$$P\approx 267\text{ W (3 s.f.)}$$
Examiner InsightCIE questions frequently give time in minutes or hours. Students who forget to convert to seconds before substituting lose the method mark and the answer mark. Exam cue: scan every given value for non-SI units before writing any equation.
Calculating Power from Energy Transferred
Key Equations
Power (from energy transferred):
$$P=\frac{\Delta E}{t}$$
Variables:
- $P$ = power, in watts (W)
- $\Delta E$ = energy transferred, in joules (J)
- $t$ = time taken, in seconds (s)
SI unit of calculated quantity: W (watt)
Rearrangements
Starting from $P=\frac{\Delta E}{t}$:
Rearranging for energy transferred:
$$\Delta E=P\times t$$
Rearranging for time:
$$t=\frac{\Delta E}{P}$$
ProportionalityPower is directly proportional to energy transferred when time is constant — doubling the energy transferred doubles the power. Power is inversely proportional to time when energy transferred is constant — doubling the time halves the power.
The equation $P=\frac{\Delta E}{t}$ is used in exactly the same way as $P=\frac{W}{t}$. The two forms exist because work done on or by an object equals the energy transferred to or from that object. Energy is transferred from one store to another — for example, from the chemical store of a fuel to the kinetic store of a vehicle — and the rate of that transfer is the power.
Choose $P=\frac{W}{t}$ when the question states a value for work done, and $P=\frac{\Delta E}{t}$ when the question states a value for energy transferred. Both give the same result because work done and energy transferred are numerically equal for the same process.
Worked Example
A 2.0 kW electric heater runs for 3.0 minutes. Calculate the energy transferred by the heater.
Converting power to SI units
$$P=2.0\times 1000$$
$$P=2000\text{ W}$$
Converting time to SI units
$$t=3.0\times 60$$
$$t=180\text{ s}$$
Finding the energy transferred
Equation used — Power (from energy transferred), rearranged for energy
$$\Delta E=P\times t$$
Given
$$P=2000\text{ W}$$
$$t=180\text{ s}$$
Substituting:
$$\Delta E=2000\times 180$$
$$\Delta E=360 000\text{ J}$$
$$\Delta E=360\text{ kJ}$$
MisconceptionStudents sometimes believe $P=\frac{W}{t}$ and $P=\frac{\Delta E}{t}$ give different answers for the same situation. They do not. Work done equals energy transferred; the two equations are alternative forms of the same relationship. Exam cue: if a question gives "energy transferred," use $\Delta E$; if it gives "work done," use $W$ — the physics is identical.
QUICK RECAP
Key Points
- Power is the rate of doing work or the rate of energy transfer.
- The SI unit of power is the watt (W): 1 W = 1 J s⁻¹.
- P = W / t gives power from work done and time.
- P = ΔE / t gives power from energy transferred and time.
- Work done equals energy transferred for the same process.
- Rearrange for work done: W = P × t.
- Rearrange for energy transferred: ΔE = P × t.
- Rearrange for time: t = W / P or t = ΔE / P.
- Power ∝ work done (at constant time).
- Power ∝ 1 / time (at constant work done).
- Doubling work done in the same time doubles power.
- Halving the time for the same work doubles power.
- Always convert minutes to seconds and kW to W before substituting.
CAN I…? PROGRESS CHECK
Self-Assessment
- Define power using CIE-approved phrasing (rate of doing work / rate of energy transfer)?
- State the SI unit of power and its equivalent in base units?
- Recall and use P = W / t, including rearrangements for W and t?
- Recall and use P = ΔE / t, including rearrangements for ΔE and t?
- Explain why P = W / t and P = ΔE / t give the same result?
- Convert minutes to seconds, kW to W, and kJ to J before substituting?
- Predict how power changes when time or work done is doubled or halved?