S2 Energy

Every change you will study in this chapter — a trolley rolling down a slope, a kettle boiling, a turbine turning — can be read as energy moving from one place or one form to another. The bookkeeping device that tells you whether a store of energy went up or down is work.

So whenever a question says "work is done", it is telling you a transfer is happening. Read it as: which store is emptying, and which store is filling? Answer that and the rest of the calculation is arithmetic.

The formula holds a trap. Speed is squared, so doubling the speed does not double the kinetic energy — it multiplies it by four. Mass enters only once, so doubling the mass simply doubles the kinetic energy. For a given mass, a fast body carries disproportionately more energy than a slow one.

That is why braking distances grow so sharply with speed, and why a car at 100 km h⁻¹ does roughly four times the damage of the same car at 50 km h⁻¹.

A delivery van has a weight of 14 000 N and travels at a steady 54.0 km h⁻¹. Find its kinetic energy. (g = 10 N kg⁻¹)

• Step 1 — find the mass. Weight is a force, not a mass. W = m g → 14 000 = m × 10 → m = 1400 kg.
• Step 2 — convert the speed. 54.0 km h⁻¹ = 54 000 m ÷ 3600 s = 15.0 m s⁻¹. (Shortcut: divide km h⁻¹ by 3.6.)
• Step 3 — substitute. K.E. = ½ × 1400 × (15.0)² = ½ × 1400 × 225 = 157 500 J = 1.58 × 10⁵ J = 158 kJ (3 s.f.).

where m is the mass in kg, g is the gravitational field strength (10 N kg⁻¹ on Earth) and h is the vertical height above a chosen level in metres.

Because the formula contains h, the value depends on where you decide zero is. Every calculation needs a reference level; the ground is the natural choice, and any object above it then has a positive G.P.E. Move the reference level and every number changes, even though nothing physical has happened.

A 0.250 kg ball is released from rest on a ledge 12.0 m above the ground and falls freely. Taking g = 10 N kg⁻¹ and ignoring air resistance: (a) what is its G.P.E. at the ledge, and (b) how fast is it moving just before impact?

• (a) G.P.E. = m g h = 0.250 × 10 × 12.0 = 30.0 J.
• (b) Falling freely, all of that store becomes kinetic energy: K.E. = 30.0 J.
  30.0 = ½ m v² = ½ × 0.250 × v², so v² = 30.0 ÷ 0.125 = 240
  v = √240 = 15.5 m s⁻¹ (3 s.f.).

Notice that the ball's mass cancelled out of the speed — something you would not see if you had gone straight to the SUVAT equations. The energy route shows you why a heavy ball and a light ball land at the same speed.

Some forms are stores that a body carries around with it; others are really transfers, energy on its way from one store to another. Both appear in exam chains, so learn to name them accurately.

Real machines are not perfectly tidy. A car engine, a fan, a bouncing ball — all of them hand part of the energy they are given to the surroundings as thermal energy, spread by friction or by the resistance of the air. Engineers call that share the wasted energy, not because it has vanished but because it has leaked into a form and a place we cannot use.

A Sankey diagram makes the split visible: one wide ribbon enters, and it divides into a useful ribbon and a wasted one, with the widths drawn in proportion to the energy each carries.

Anything that is moving is carrying energy. A loaded lorry rolling along a highway, a cyclist freewheeling downhill and a cricket ball crossing the pitch all store it, and the amount each one holds depends on how much matter is on the move and how quickly that matter travels.

Speed is squared in that formula; mass is not. Doubling the speed multiplies the kinetic energy by four, while doubling the mass only doubles it, and halving the speed leaves a quarter of the original energy. That is why a vehicle travelling twice as fast hands its brakes four times as much energy to remove. A speed quoted in km/h must be converted by dividing by 3.6 before it goes anywhere near the calculation.

Lift a book onto a high shelf and you have transferred energy to it. That energy sits there stored, waiting to be released the moment the book is knocked off. What matters is how far the body has been raised above a chosen level and how strong the pull of gravity is where it is raised.

The three quantities multiply together, so doubling any one of them — mass, field strength or height — doubles the energy stored. Near the Earth's surface g is taken as 10 N/kg unless a question states otherwise.

G.P.E. only has meaning against a reference level, which we choose and label zero — usually the ground or the bench top, so that any body above it holds positive G.P.E. Choosing a different zero shifts every value by the same fixed amount, and never changes the change in energy between two positions, which is what does useful work.

Energy rarely stays in one form. It passes from one store to another, and marks are earned by naming each step in order, from the starting form to the final one. Continue the ball from Worked example 2 and the chain is short: as it falls its store of gravitational potential energy shrinks while its kinetic energy grows, and the total is unchanged. If all 72 J becomes kinetic energy by the time the ball lands, then:

½ mv² = 72 J
½ × 0.60 × v² = 72
0.30 v² = 72
v² = 240
v = 15.5 m/s (3 s.f.)

The height told you how much energy was available; conservation tells you what the falling body can do with it. Pairing the two formulas in this way is the most common structure in questions on this topic.

Doing work on something and transferring energy to it are two ways of describing the same event. A force that pushes or pulls a body through a distance is transferring energy to that body, and the size of the transfer is what we call the work done.

Direction matters. Only the part of the distance covered along the line of the force counts towards the work done; push downwards on a box that slides sideways and you transfer no energy to it at all.

• A force doing work on a body increases the energy stored in that body by exactly the work done.
• A body doing work on something else gives up that same amount of energy, and its own store falls.
• Push a trolley along a level floor and you transfer energy to it, but if it moves at a steady speed against friction, that energy leaves again as thermal energy almost at once.

Lifting links this section to the last. The upward force needed to raise a load steadily equals its weight, mg, and the distance moved is the vertical rise h, so W = mg × h. That is identical to the gain in G.P.E., which is why the two formulas can be used interchangeably for a lift.

No real device returns every joule handed to it. Friction in moving parts, stirring of air and the generation of sound all divert a share of the input into thermal energy, which spreads into the surroundings and cannot easily be gathered up again. A machine is judged by how much of its input becomes the output we actually want.

Powers can replace energies in that fraction, because power is energy transferred per second: useful power output divided by total power input gives the same number. An efficiency of 0.571 and one of 57.1% mean exactly the same thing.

Read that as a statement about the machine: for every 100 J put in, about 57 J ends up stored as G.P.E. in the crate and about 43 J heats the winch and the air around it. Oiling the moving parts raises the efficiency, but 100% is unreachable — no real mechanism is frictionless.

Every appliance we plug in or fuel we burn is fed by something dug up, pumped, grown or caught. To sort those supplies, ask one question: if we keep using this, will nature put it back as fast as we take it?

A non-renewable fuel is also a stored chemical store we release by burning it; most renewables arrive as a flow, so we must capture them while they pass.

However different the sources look, the machinery downstream is remarkably uniform: a power station is essentially a device for making a shaft spin steadily, then turning that spin into electrical energy.

• Make a fluid hot, or fast, or high up. Burning a fuel, splitting nuclei, focusing sunlight or tapping the Earth's interior yields heat; falling water carries gravitational potential energy; moving air carries kinetic energy.
• Rotate a turbine, then spin a generator. Expanding steam, rushing water or blowing air turns angled blades on a shaft; coils turning in a magnetic field then induce a current, so the shaft's kinetic energy becomes electrical energy.
• Step up, transmit, step down. Transformers raise the voltage for long-distance transmission and lower it again before the supply enters homes.

As chains of conversions, the common routes are:

• Coal, oil or gas: chemical potential energy → thermal energy → kinetic energy → electrical energy
• Nuclear fuel: nuclear energy → thermal energy → kinetic energy → electrical energy
• Reservoir hydroelectricity: gravitational potential energy of stored water → kinetic energy → electrical energy
• Geothermal plant: thermal energy of underground steam → kinetic energy → electrical energy
• Wind farm: kinetic energy of air → kinetic energy of the turbine → electrical energy

Solar photovoltaic is the outlier: no turbine is involved, because light falling on a semiconductor cell produces a current directly.

The table groups the mains by what is captured, how it reaches the grid, and what it costs us.

Fossil fuels are the buried remains of living things, transformed by heat and pressure over millions of years. Being concentrated stores, a small mass carries a large amount of chemical potential energy, so they are easy to transport and burn on demand. The cost is in the exhaust: combustion adds carbon dioxide, sulphur and nitrogen compounds and fine particles, linked to acid deposition and respiratory illness.

Nuclear fuel reaches heat by another route: uranium nuclei are split in a reactor, and the energy released warms a coolant that raises steam for the turbines. The station emits no carbon dioxide, which is why countries with few fuel reserves treat it as a way to secure supply. Against that: spent fuel stays dangerously radioactive for thousands of years; reactors and their decommissioning are capital-intensive; and an accident can render a large area uninhabitable for decades.

Environmental. Renewables in operation add little or no carbon dioxide, so choosing them lowers the emissions and the pollutants that damage air quality. Their impacts are local — a flooded valley, a changed estuary, turbines on a ridge. Non-renewable fuels reverse the pattern: everyday operation is the global problem, while their worst impacts are rarer.

Practical. Fossil fuels are dense, cheap to move, storable in a tank and available whenever the grid calls, which makes them the dispatchable backbone many systems still rely on; nuclear gives the same steady output without the emissions, if a country can afford it and accept the waste. Renewables invert that profile: no fuel to buy, but high upfront cost, output tied to weather, land to find, and a grid built for a few large stations rather than thousands of small ones. Storage — batteries, pumped hydro, stored heat — is the missing piece that turns an intermittent supply into a dependable one.

Meters and bills do not count joules. They count kilowatt-hours (kWh) — the energy transferred by one kilowatt of power running for one hour. A kilowatt is 1 000 watts, so

A bill multiplies that energy by the tariff, quoted in cents per kWh, and adds a fixed charge for being connected, used or not.

A worked figure of our own. A household runs a 1.2 kW air-conditioner for 5 hours every day. What does it add to the bill for a 30-day month, at a tariff of 28.0 cents per kWh?

• Each day: E = P × t = 1.2 kW × 5 h = 6.0 kWh
• For the month: 6.0 kWh × 30 = 180 kWh
• The same energy in joules: 180 × 3.6 MJ = 648 MJ
• Cost: 180 kWh × $0.280 per kWh = $50.40

Two habits follow. Because the tariff multiplies energy, not power, a low-powered appliance left on for hours can cost more than a high-powered one used briefly — a 100 W fan running ten hours uses the same 1.0 kWh as a 2 kW kettle boiling for half an hour.

Two habits cover most of the list. First, write the formula, then substitute, then give the unit with the number. Second, check that every quantity carries the unit the formula expects — kg, m, s, N — because a unit that looks wrong usually is wrong.

Descriptions are marked on the forms named and the order given, not on the length of the sentence. Follow three rules.

1. Name the form at each stage, with its direction. Use the syllabus vocabulary: kinetic energy, gravitational potential energy, elastic potential energy, chemical potential energy, thermal energy, light energy, sound energy, electrical energy, nuclear energy. Write the chain with arrows rather than prose, and include every conversion the question expects to see.
2. Start from the source and end at the final form asked for. If the question asks how a power station produces electricity, your chain must finish at electrical energy; stopping at thermal energy loses the last mark.
3. Use the word "converted" or "transferred", not "used up". Energy is never used up. In a real device some of it is transferred to the surroundings as thermal energy — say that explicitly when the question asks about wasted energy, and name the cause, usually friction or air resistance.

A model chain for a falling ball as it drops and lands: gravitational potential energy → kinetic energy (during the fall) → thermal energy and sound energy (on impact). Note how the chain records what happens at each stage, in order, and does not claim the total has fallen.

The wording of the question tells you which relationship to reach for. Match the phrase in the stem to the row below.

If a question gives a mass and a height and asks about work done, remember the two views agree: lifting slowly, the force applied equals the weight, so W = mgh and it equals the gain in gravitational potential energy. Quote either route — show the substitution.

Question. A student lifts a box of mass 12.0 kg from the ground onto a shelf 1.5 m above the ground, then lets it slide down a smooth ramp back to the ground. Take g = 10 N kg⁻¹.

1. State the SI unit of energy and express 1 kJ in joules. [1]
2. Calculate the gravitational potential energy gained by the box as it is lifted onto the shelf. [2]
3. The student takes 3.0 s to lift the box. Calculate the average power delivered. [2]
4. Calculate the speed of the box when it reaches the bottom of the ramp, assuming no energy is transferred to the surroundings. [3]
5. In practice the ramp is not smooth and the box reaches the ground more slowly. Explain this result in terms of energy. [2]

1. The joule (J); 1 kJ = 1 000 J. (1 mark — both parts needed.)
2. G.P.E. = mgh
   = 12.0 × 10 × 1.5
   = 180 J
   (2 marks — formula or correct substitution, then the answer with its unit.)
3. P = W / t
   = 180 / 3.0
   = 60 W
   (2 marks — the work done equals the energy transferred, and the unit must be the watt.)
4. Energy is conserved, so the gravitational potential energy at the top is transferred to kinetic energy at the bottom:
   ½ mv² = mgh = 180 J
   ½ × 12.0 × v² = 180
   v² = 30
   v = 5.5 m s⁻¹ (2 s.f.)
   (3 marks — the statement that the energy is transferred, the rearranged substitution, and the answer with its unit.)
5. Friction between the box and the ramp transfers some of the box's energy to the surroundings as thermal energy. The total energy is still conserved, but the useful kinetic energy at the bottom is smaller, so the speed is lower than the value calculated in (d). (2 marks — naming thermal energy as the wasted form, and stating that the total energy is unchanged.)