Power
Work, Energy & Power - OCR A-Level Physics
Key Definition
Power
The rate at which work is done, or equivalently the rate of energy transfer. SI unit: the watt (W), where $1 \text{ W} = 1 \text{ J s}^{-1}$.
The rate at which work is done, or equivalently the rate of energy transfer. SI unit: the watt (W), where $1 \text{ W} = 1 \text{ J s}^{-1}$.
$$P = \frac{W}{t} = \frac{\Delta E}{t}$$
Power and velocity
For an object moving at constant velocity $v$ under a constant force $F$, the power delivered by that force is $P = Fv$. This follows from $P = W/t = Fx/t = F \cdot (x/t) = Fv$.
$$P = Fv$$
- At constant velocity, by Newton's first law the driving force equals the total resistive force. So the engine power equals the resistive force times speed: $P_{\text{engine}} = F_{\text{resistive}} \times v$.
- At maximum speed of a vehicle, all the engine power is used to overcome resistive forces: $P_{\max} = F_{\text{resistive}} \times v_{\max}$.
- If engine power is fixed, increasing speed must decrease the driving force. This is why a car accelerates less and less as speed grows, even at full throttle.
- Useful order of magnitude: a person walking briskly delivers about $100 \text{ W}$; a small electric kettle is about $2 \text{ kW}$; a family car at motorway speed needs about $20 \text{ kW}$.
Common Mistake
MEDIUM
Wrong: Using $P = Fv$ with the resultant force during acceleration, instead of with the driving force at constant velocity.
Right: $P = Fv$ gives the power delivered by the force $F$ at the instantaneous speed $v$. For the engine's output power at constant velocity, use the driving (engine) force; for power dissipated against drag, use the drag force.
Right: $P = Fv$ gives the power delivered by the force $F$ at the instantaneous speed $v$. For the engine's output power at constant velocity, use the driving (engine) force; for power dissipated against drag, use the drag force.
Worked Example [3 marks]
A car has a maximum power output of $45 \text{ kW}$ and experiences a total resistive force of $1500 \text{ N}$ at high speeds. Calculate its maximum speed.
Show Solution
1
At maximum speed, driving force = resistive force
$F = 1500 \text{ N}$ (constant velocity, so resultant force is zero).
[1]2
$P = Fv$, so $v = \frac{P}{F} = \frac{45\,000}{1500}$.
[1]3
$v_{\max} = 30 \text{ m s}^{-1}$.
[1]Answer
$30 \text{ m s}^{-1}$
Examiner Tips and Tricks
- If a question gives a power and asks for time or energy, choose between $P = W/t$ and $P = Fv$ based on what data you have. Both are on the data sheet.
- Always convert $\text{kW}$ to $\text{W}$ and $\text{km h}^{-1}$ to $\text{m s}^{-1}$ before substituting.