Work done by a force, kinetic and potential energy, the work-energy principle, the conservation of mechanical energy, and power as the rate of doing work.

What this dot point is asking

Edexcel Further Mechanics wants you to compute work done by a force, use kinetic and gravitational potential energy, apply the work-energy principle, use conservation of mechanical energy where only conservative forces act, and relate power to force and velocity. Power problems for vehicles climbing slopes against resistance are a recurring exam favourite.

Work and energy

Work is the transfer of energy by a force as its point of application moves. Only the component of the force along the displacement does work, hence the $\cos\theta$ factor; a force perpendicular to the motion (such as a normal reaction) does no work. Kinetic energy and gravitational potential energy are the two mechanical energy stores at this level.

The work-energy principle

The net work done by all forces on a body equals its change in kinetic energy. This is often the cleanest route when forces vary or when you do not need the time taken, only the speeds and distances. When the only forces doing work are conservative (gravity, ideal springs), mechanical energy is conserved and you can equate total energy at two points. When friction or air resistance acts, include the work done against those forces explicitly.

Power

Power is the rate at which work is done, measured in watts ($\text{W}$). For a force driving motion in the direction of travel, power equals force times speed, $P = Fv$. This relationship explains why a vehicle at constant engine power accelerates less as it speeds up: the driving force $F = \frac{P}{v}$ falls as $v$ rises, until at maximum speed the driving force just balances the total resistance.

Examples in context

Work, energy and power underpin much of mechanics. Conservation of energy is the standard tool for finding speeds in a vertical circle (the circular-motion dot point) and for projectile or slope problems where time is not needed. The kinetic energy lost in a collision, computed in the elastic-collisions dot point, is exactly the mechanical energy not conserved when $e < 1$. The integral definition of work, $W = \int F\,dx$ for a variable force, connects to further calculus, as does the work done by a spring, $\frac{1}{2}kx^2$. Power-speed problems for cars and trains on inclines are a classic application combining all three concepts.

Try this

Q1. A car engine works at $12\,000\,\text{W}$, driving the car at $20\,\text{m s}^{-1}$. Find the driving force. [2 marks]

• Cue. $F = \frac{P}{v} = \frac{12000}{20} = 600\,\text{N}$.

Q2. Find the kinetic energy of a $3\,\text{kg}$ mass moving at $4\,\text{m s}^{-1}$. [1 mark]

• Cue. $\operatorname{KE} = \frac{1}{2}(3)(4^2) = 24\,\text{J}$.

Q3. A $1\,\text{kg}$ ball is dropped from $5\,\text{m}$. Use energy conservation to find its speed on landing (ignore air resistance, $g = 9.8$). [2 marks]

• Cue. $\frac{1}{2}v^2 = gh$, so $v^2 = 2 \times 9.8 \times 5 = 98$ and $v \approx 9.9\,\text{m s}^{-1}$.