What is the law of conservation of energy?
Energy is the ability to do work. It is measured in joules (J).
The law says: energy cannot be created or destroyed. It can only change form or move from one object to another.
So in a closed system (nothing goes in or out), the total energy is always the same. A closed system is like a sealed box: we count everything inside it.
Forms of energy you will meet: potential (stored by height or stretch), kinetic (movement), thermal (heat), chemical (in fuel and food), electrical, light and sound.
Mechanical energy: potential and kinetic
Potential energy PE = m g h (mass × 9.8 m/s² × height).
Kinetic energy KE = ½ m v² (half × mass × speed²).
Mechanical energy = PE + KE.
If there is no friction or air resistance, mechanical energy stays constant:
m g h₁ + ½ m v₁² = m g h₂ + ½ m v₂²
A falling ball, a pendulum and a roller coaster all trade PE for KE and back again. The mass cancels, so the speed at the bottom is v = √(2gh) for any mass.
Mechanical energy into heat: friction
Rub your hands fast. They get warm. That warmth came from the movement energy of your hands.
Friction turns mechanical energy into thermal energy (heat). Mechanical energy goes down, but the energy is not lost. It is now in the warmer surfaces and air.
Energy check: mechanical energy at start = mechanical energy at end + heat produced.
Examples: brakes of a bicycle get hot; a meteor glows in the air; a drill bit heats up.
Energy conservation in thermal processes and fuel burning
When hot and cold things touch, heat flows from hot to cold until they reach the same temperature. Heat given by the hot body = heat taken by the cold body (if none escapes).
Heat needed to warm something: Q = m c ΔT (mass × specific heat capacity × change in temperature). For water, c ≈ 4200 J/(kg·°C).
Burning fuel releases chemical energy: Q = q m, where q is the specific heat of combustion (energy per kg of fuel). For example, LPG gives about 46 MJ per kg.
On a real stove, not all the fuel's heat reaches the water. Some warms the pot and the air. Energy is still conserved; it just goes to places we did not want.
Efficiency and energy flow diagrams
Efficiency = useful energy output ÷ total energy input × 100%.
No real machine is 100% efficient, because some energy always becomes heat or sound we cannot use. An LED bulb turns about 40% of its electrical energy into light; an old filament bulb only about 5%.
An energy flow (Sankey) diagram shows this with arrows: a wide arrow in, a useful arrow out, a wasted arrow bending away. The widths always add up to the input.
Try it at home
Bouncing ball test: drop a ball from 1 m next to a wall. Mark how high it bounces. Predict first: will it come back to 1 m? It will not. Where did the missing energy go? (Sound of the bounce, heat in the ball and floor.) Then try the sliders in the 3D above and see the heat bar grow.
Key formulas and definitions
- PE = m g h
- KE = ½ m v²
- PE₁ + KE₁ = PE₂ + KE₂ (no friction)
- Mechanical energy at start = mechanical energy at end + heat
- Q = m c ΔT
- Q = q m (burning fuel)
- Efficiency = useful output ÷ total input × 100%
Worked examples
1. A 2 kg ball is held 5 m above the ground. Find its potential energy (g = 9.8 m/s²).
PE = m g h = 2 × 9.8 × 5 = 98 J.
2. The ball from Example 1 is dropped. Find its speed just before it hits the ground (no air resistance).
All PE becomes KE: 98 = ½ × 2 × v². So v² = 98, v ≈ 9.9 m/s. (Or v = √(2gh) = √(98) ≈ 9.9 m/s.)
3. A 50 kg child slides down a 3 m high slide and reaches the bottom at 6 m/s. How much energy became heat?
PE at top = 50 × 9.8 × 3 = 1470 J. KE at bottom = ½ × 50 × 36 = 900 J. Heat = 1470 − 900 = 570 J.
4. How much heat is needed to warm 2 kg of water from 20 °C to 70 °C? (c = 4200 J/kg°C)
Q = m c ΔT = 2 × 4200 × 50 = 420 000 J = 420 kJ.
5. Burning 0.02 kg of LPG (q = 46 MJ/kg) heats the water in Example 4. What is the efficiency of the stove?
Energy from fuel = 0.02 × 46 000 000 = 920 000 J. Useful = 420 000 J. Efficiency = 420 000 ÷ 920 000 × 100% ≈ 46%.
6. A motor gets 500 J of electrical energy and lifts a 10 kg box by 4 m. Find its efficiency.
Useful = m g h = 10 × 9.8 × 4 = 392 J. Efficiency = 392 ÷ 500 × 100% = 78.4%.
Common mistakes
- Saying energy is 'lost' or 'used up'. It is not destroyed; it changes into a less useful form, usually heat.
- Forgetting to square the speed in KE = ½ m v². Doubling speed makes KE four times bigger.
- Using mechanical energy conservation when there is friction. Then you must add the heat term.
- Writing efficiency more than 100%. Useful output can never be bigger than input.