Work, Energy and Power
FoundationWork is the energy transferred by a force along a displacement, W = Fs cosθ; kinetic and gravitational potential energy are calculated with ½mv² and mgh, conservation of energy links the conversions, and power P = W/t measures how fast work is done, in watts.
- Work is energy transferred by a force along a displacement: W = Fs cosθ, in joules; a force perpendicular to the displacement does no work
- Kinetic energy KE = ½mv²; gravitational potential energy PE = mgh; doubling the speed quadruples the kinetic energy
- Conservation of energy: energy only changes form — resistance turns mechanical energy into internal energy
文體小知識：科學知識說明文——講解功、動能、重力位能與功率的定義及公式，說明能量守恆怎樣把力與能量變化連繫起來，是力學能量部分的第一課。
§Text Passage
1Physics defines work as the energy transferred by a force acting along a displacement. When a force acts on an object and the object moves in the direction of the force, the force does work, measured in joules. Two conditions must hold: there must be a force, and there must be a displacement along that force. For example, carrying a school bag while walking along a level road, the supporting force is vertical and the displacement is horizontal, so the supporting force does no work.
2The two most common forms of energy in mechanics are kinetic energy and gravitational potential energy. Kinetic energy is the energy an object has because of its motion, equal to half the mass times the square of the speed; gravitational potential energy is the energy stored because of height, equal to mass times gravitational acceleration times height. For example, as a cable car climbs at constant speed its kinetic energy stays the same while its potential energy grows, both supplied by the electrical energy of the motor.
3The law of conservation of energy states that energy is never created or destroyed — it only changes from one form to another, with the total unchanged. As an object falls, potential energy turns into kinetic energy and the speed grows; with friction or air resistance, part of the mechanical energy becomes internal energy and the object warms up slightly. Writing out an energy chain is the standard way to explain machines and everyday phenomena.
4Power measures how fast work is done: it is defined as the energy transferred per second, equal to work divided by time, and measured in watts — one watt means one joule per second. For example, a vacuum cleaner rated at 600 W converts 600 J of electrical energy into kinetic energy, sound and internal energy every second; running for 10 minutes, it uses 360000 J. The larger the power, the faster the same work is done.
5S5 mechanics often combines work, energy and power in one question. Draw the directions of the force and the displacement, find the angle, choose the right formula, and finish with the unit. For example, pulling a suitcase 20 m with a 250 N force at 30° to the ground: only the component along the displacement does work. This is a classic DSE Paper 1 setup, and both steps and units are indispensable.
1Summary 重點摘要
Work is the energy transferred by a force along a displacement, W = Fs cosθ; kinetic and gravitational potential energy are calculated with ½mv² and mgh, conservation of energy links the conversions, and power P = W/t measures how fast work is done, in watts.
2Vocabulary cards Vocabulary
The energy transferred by a force along a displacement; W = Fs cosθ, measured in joules.
The ability to do work; energy only changes form and is never created or destroyed.
The energy an object possesses due to its motion, KE = ½mv².
The energy stored in an object because of its height, PE = mgh.
The energy transferred per second, P = W/t; measured in watts.
The unit of power; 1 watt equals 1 joule per second.
The unit of energy and work: 1 J is the work done by a 1 N force over 1 m along the displacement.
Energy is neither created nor destroyed, only converted from one form to another.
The projection of a force along a direction; only the component along the displacement does work.
The process of energy changing from one form to another, e.g. electrical energy into kinetic and internal energy.
3Step by step Steps
- Step 1: The definition and two conditions of workWork measures energy transfer: when a force acts on an object and the object moves along the direction of the force, that force does work, measured in joules. Two conditions must hold at the same time: there must be a force, and a displacement along it. For example, pushing a shopping trolley forward at a supermarket, the pushing force does work; but holding a suitcase while standing still, though your arms strain, the suitcase does not move and the supporting force does zero work. Similarly, swinging a bucket on a rope in a horizontal circle, the tension points at the centre, perpendicular to the motion, so it does no work either.
- Step 2: When force and displacement are at an angle: W = Fs cosθWhen the force F makes an angle θ with the displacement s, the formula is W = Fs cosθ: only the component F cosθ along the displacement does work. At θ = 0°, cosθ = 1 and the work is largest; at θ = 90°, cosθ = 0 and the work is zero. For example, pulling a suitcase 20 m with a 250 N force at 30° to the ground gives W = 250 × 20 × cos30° = 5000 × 0.866 ≈ 4330 J. If instead you carry the suitcase upright and walk 20 m, the angle is 90° and the work done is zero. The cos button on a calculator gives the cosine directly, and the answer must carry the unit joule.
- Step 3: Kinetic energy KE = ½mv²Kinetic energy is the energy an object has because of its motion: KE = ½mv², with m in kg and v in m s⁻¹, still in joules. Note the square: double the speed and the kinetic energy quadruples. For example, an 800 kg car at 10 m s⁻¹ has KE = ½ × 800 × 10² = 40000 J; at 20 m s⁻¹ it is 160000 J, four times as much. This is why speeding is so dangerous: the kinetic energy to be dissipated when braking grows with the square of the speed, so the braking distance lengthens sharply.
- Step 4: Gravitational potential energy PE = mghGravitational potential energy is the energy stored because of height: PE = mgh, where h is the height above a chosen reference level and g = 9.8 m s⁻². The greater the mass or the height, the more energy is stored. For example, lifting a 0.5 kg bottle of water from a desk to a 2 m cupboard top raises its PE by 0.5 × 9.8 × 2 = 9.8 J; a 60 kg student walking up to a classroom 5 m above the ground gains 60 × 9.8 × 5 = 2940 J — the bulk of the energy cost of climbing, and a good reason stairs are tiring.
- Step 5: Conservation of energy and energy chainsThe law of conservation of energy states that energy is never created or destroyed, only converted from one form to another, with the total unchanged. In mechanics, kinetic and potential energy keep exchanging: a falling object turns potential energy into kinetic energy and speeds up; a ball thrown upward turns kinetic energy into potential energy and slows down. Without resistance the total stays constant. For example, a 0.2 kg apple dropping from a 3 m tree arrives with kinetic energy roughly equal to the lost PE, 0.2 × 9.8 × 3 = 5.88 J; with air resistance some energy becomes internal energy, yet the total is still conserved.
- Step 6: Power P = W/tPower measures how fast work is done or energy is transferred: it is the energy transferred per second, P = W/t, in watts, with 1 W = 1 J s⁻¹. The larger the power, the shorter the time for the same work. For example, transferring 600 J in 2 s takes a power of 300 W; a 600 W vacuum cleaner running for 10 minutes — 600 seconds — uses E = Pt = 600 × 600 = 360000 J, i.e. 3.6 × 10⁵ J. Convert the time into seconds before substituting, otherwise the result is out by a factor of sixty — one of the most common slips in calculations.
- Step 7: Estimating human power outputThe power formula can be rearranged to find time, or to rate machines and the human body. An average person sustains one to two hundred watts during exercise, with short bursts of several hundred. For example, a 60 kg student walking up a 3 m staircase in 10 s does work against gravity of W = mgh = 60 × 9.8 × 3 = 1764 J, giving a power of P = 1764 ÷ 10 ≈ 176 W; running up in 5 s nearly doubles the power to about 353 W. The work done is the same either way — only the speed differs, which is exactly what power measures: the rate of doing work.
4Diagrams Diagrams
5Worked example Example
Example: a traveller pulls a suitcase with a 250 N force at 30° to the horizontal ground, moving it 20 m along the ground. (a) Find the work done by the pulling force. (cos 30° = 0.866) (b) Back home, the traveller vacuums for 10 minutes with a 600 W vacuum cleaner; find the electrical energy used, in joules.
- (a) Write the formula: W = Fs cosθ, with F = 250 N, s = 20 m, θ = 30°
- (a) Substitute: W = 250 × 20 × 0.866
- (a) Calculate: W ≈ 4330 J
- (b) Vacuum cleaner: t = 10 × 60 = 600 s, so E = Pt = 600 × 600
- (b) Calculate: E = 360000 J = 3.6 × 10⁵ J
- Answer: (a) W = Fs cosθ = 250 × 20 × 0.866 ≈ 4330 J. (b) E = Pt = 600 × 600 = 360000 J, i.e. 3.6 × 10⁵ J in ten minutes. Note that (a) needs the factor cos30° because the force is at an angle to the displacement, and (b) needs the time converted to 600 seconds first — both common places to lose marks.
6Common mistakes Mistakes
7Quick check Quick check
Q1What are the two conditions for doing work? Give an example where no work is done.
Q2A horizontal force of 50 N pushes a box 8 m. How much work is done?
Q3A 1000 kg car speeds up from 10 m s⁻¹ to 20 m s⁻¹. By how much does its kinetic energy increase?
Q4A 600 W fan runs for 5 minutes. How much electrical energy does it use?
Q5As an apple falls from a tree to just before impact, how does energy convert?
8Recap Recap
- Work is energy transferred by a force along a displacement: W = Fs cosθ, in joules; a force perpendicular to the displacement does no work
- Kinetic energy KE = ½mv²; gravitational potential energy PE = mgh; doubling the speed quadruples the kinetic energy
- Conservation of energy: energy only changes form — resistance turns mechanical energy into internal energy
- Power P = W/t measures the rate of doing work, 1 W = 1 J s⁻¹; convert time to seconds before substituting
- Four steps to solve: draw the directions, find the angle, substitute, write the unit