Power: Definition, P = Fv, and Problem Solving (9709 Mechanics 4.5) — Mathematics

Introduction

Power is the rate at which a force does work. In the 9709 Mechanics exam, power questions typically involve vehicles (cars, lorries, trains) moving along roads or hills, and require you to connect the engine's driving force to the vehicle's speed and acceleration. The key formula $P = Fv$ is one of the most frequently applied in the Mechanics paper, appearing in problems that combine Newton's Second Law with energy concepts.

Core Concept

Work Done (Prerequisite Recap)

When a constant force $F$ moves its point of application a distance $d$ in the direction of the force, the work done is:

W = Fd

Power as Rate of Doing Work

Power is the rate at which work is done:

P = \frac{\text{Work done}}{\text{Time taken}} = \frac{W}{t}

This gives average power when work $W$ is done over a time interval $t$ .

Instantaneous Power: $P = Fv$

For a force $F$ acting in the direction of motion at a velocity $v$ :

P = Fv

Derivation: If the force moves through a small distance $\delta d$ in time $\delta t$ , then:

P = \lim_{\delta t \to 0} \frac{F\,\delta d}{\delta t} = F\,\frac{dd}{dt} = Fv

This is the instantaneous power — valid at any specific moment when the velocity is $v$ .

Units

The SI unit of power is the watt (W), where $1\,\text{W} = 1\,\text{J\,s}^{-1} = 1\,\text{N\,m\,s}^{-1}$ .

Problems typically use kilowatts (kW), so always convert: $1\,\text{kW} = 1000\,\text{W}$ .

Driving Force from Power

Rearranging $P = Fv$ :

F = \frac{P}{v}

This is the driving force (engine thrust) at speed $v$ when the engine works at power $P$ . As speed increases, the driving force decreases (for constant power output).

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