Energy, work and power
Introduced| English | Meaning |
|---|---|
| power/ˈpaʊə/ | The rate of energy transfer or work done. |
| potential energy/pəˈtenʃl ˈenədʒi/ | Energy stored because of position or configuration. |
| kinetic energy/kɪˈnetɪk ˈenədʒi/ | The energy an object has because of its motion. |
| conservation of energy/ˌkɒnsəˈveɪʃn ɒv ˈenədʒi/ | The principle that energy cannot be created or destroyed, only transferred or transformed. |
| work done/wɜːk dʌn/ | Energy transferred when a force acts through a displacement, calculated using the component of force along that displacement. |
| joule/dʒuːl/ | The SI unit of energy and work. |
| gravitational potential energy/ˌɡrævɪˈteɪʃənl pəˈtenʃl ˈenədʒi/ | Energy stored because of an object’s position in a gravitational field. |
| watt/wɒt/ | The SI unit of power, equal to one joule per second. |
The energy that powers 功率 everything
- A roller coaster at the top of its first hill has maximum potential energy 势能. As it plunges down, that energy converts to kinetic energy 动能 — speed.
- Conservation of energy 能量守恒 lets you predict the speed at any point without knowing the path in between.
Work done 功
- Work done by a constant force: $W = Fd\cos\theta$ (measured in joules 焦耳).
- Work is the energy transferred by a force acting through a distance.
Worked example. A 50 N force pulls a box 4 m along the floor at $30^{\circ}$ to the horizontal. $W = 50 \times 4 \times \cos 30^{\circ} = 200 \times 0.866 = 173.2\text{ J}$.
A roller coaster trades potential energy for kinetic energy
A 50 N force pulls a box 4 m along the floor (θ = 0°). Work done = Fd cos θ. Find it (J).
Difficulty 2/5 · 140 coins + a small bonus on first completion today
W = 50 × 4 × cos 0° = 50 × 4 × 1 = 200 J.
Match each quantity to its formula or meaning.
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Energy is conserved: KE and PE swap, while power is the rate of doing work.
Kinetic and potential energy
- Kinetic energy (movement): $\text{KE} = \tfrac12 mv^2$.
- Gravitational potential energy 重力势能 (height): $\text{PE} = mgh$.
KE depends on $v^2$, not $v$. Doubling the speed quadruples the kinetic energy. A car at 60 mph has four times the KE of one at 30 mph — and needs four times the braking distance.
Conservation of energy
PE + KE = constant
As the ball falls, potential energy turns into kinetic energy — but the total never changes.
What is the kinetic energy of a 2 kg object moving at 3 m/s (KE = ½mv²), in J?
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KE = ½ × 2 × 3² = ½ × 2 × 9 = 9 J.
Taking g = 10, what is the PE of a 2 kg object raised 5 m (PE = mgh), in J?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
PE = mgh = 2 × 10 × 5 = 100 J.
Doubling the speed of an object doubles its kinetic energy.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
KE = ½mv², so doubling v quadruples KE (2² = 4 times).
Conservation of energy
- With no friction, total energy is conserved (conservation of energy).
- Loss in PE = gain in KE (for a falling object): $mgh = \tfrac{1}{2}mv^2$.
A 1 kg ball falls from 5 m (g = 10). Using mgh = ½mv², find the speed at the bottom (m/s).
Difficulty 2/5 · 140 coins + a small bonus on first completion today
mgh = ½mv² → v² = 2gh = 2(10)(5) = 100 → v = 10 m/s.
Power
- Power is the rate of doing work (in watts 瓦特): $P = \dfrac{W}{t}$.
- For a force pulling along the motion: $P = Fv$.
- Work is the scalar product of force and displacement; differentiating velocity gives the instantaneous acceleration.
A car works at 12 kW at 20 m/s. Using P = Fv, what is the driving force, in N?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
F = P/v = 12000/20 = 600 N.
You've got it
- work $W = Fd\cos\theta$; KE $= \tfrac12 mv^2$; PE $= mgh$
- with no friction, total energy is conserved
- power = rate of work; $P = Fv$