Energy, work and power
| English | Chinese | Pinyin |
|---|---|---|
| power | 功率 | gōng lǜ |
| potential energy | 势能 | shì néng |
| kinetic energy | 动能 | dòng néng |
| conservation of energy | 能量守恒 | néng liàng shǒu héng |
| work done | 功 | gōng |
| joule | 焦耳 | jiāo ěr |
| gravitational potential energy | 重力势能 | zhòng lì shì néng |
| watt | 瓦特 | wǎ tè |
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).
W = 50 × 4 × cos 0° = 50 × 4 × 1 = 200 J.
Match each quantity to its formula or meaning.
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?
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?
PE = mgh = 2 × 10 × 5 = 100 J.
Doubling the speed of an object doubles its kinetic energy.
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).
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?
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; $P = Fv$