---
title: "Energy: Forms and Conservation"
book: "High School Physics"
subject: physics
language: en
chapter: 9
exercises: 15
source: https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation
---

# Chapter 9 — Energy: Forms and Conservation

A moving car, a charged phone, a hot oven and a raised hammer share one property: each can make something happen. Physics measures that capacity with a single number — [energy](#def-g10-energy-conservation-energy) — and keeps its accounts under one iron rule: the total never changes, it only moves and changes costume. This chapter opens the ledger: the forms [energy](#def-g10-energy-conservation-energy) takes, the chains it travels, and what a [kilowatt-hour](#def-g10-energy-conservation-kwh) buys.

## 9.1 Energy: the common currency

**Definition 9.1 (Energy and the joule).**

*Energy* measures a system’s capacity to produce changes: set matter in motion, lift it, heat it, light it. Its [unit](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) is the *joule* ($\mathrm{J}$). Stored, transferred, converted — every branch of physics counts in this one currency.

**Definition 9.2 (Kinetic energy).**

A body of mass $m$ (in $\mathrm{kg}$) moving at speed $v$ (in $\mathrm{m}/\mathrm{s}$) carries the *kinetic energy* $E_k = \tfrac12\, m v^2$: doubling the mass doubles it, doubling the speed quadruples it.

**Example 9.3 (Car versus pedestrian).**

A $1300\,\mathrm{kg}$ car at $100\,\mathrm{km}/\mathrm{h}$ $= 27.8\,\mathrm{m}/\mathrm{s}$ carries $E_k = \tfrac12 \times 1300 \times 27.8^2 \approx 5.0 \times 10^{5}\,\mathrm{J}$; a $70\,\mathrm{kg}$ pedestrian at $1.5\,\mathrm{m}/\mathrm{s}$, $\tfrac12 \times 70 \times 1.5^2 \approx 79\,\mathrm{J}$ — six thousand times less. Speed is expensive: the square sees to it.

**Definition 9.4 (Gravitational potential energy).**

A body of mass $m$ at height $h$ (in $\mathrm{m}$) above a chosen reference level stores the *gravitational potential energy* $E_p = m g h$, with $g = 9.81\,\mathrm{N}/\mathrm{kg}$ — valid near the ground, where $g$ is effectively constant. Only *differences* of height matter: the level where $E_p = 0$ is chosen freely.

**Example 9.5 (Water tower).**

Each cubic [metre](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) ($1000\,\mathrm{kg}$) of water stored $30\,\mathrm{m}$ up a water tower holds $1000 \times 9.81 \times 30 \approx 2.9 \times 10^{5}\,\mathrm{J}$ — about six tenths of the car’s [kinetic energy](#def-g10-energy-conservation-kinetic) above. Cities stockpile [energy](#def-g10-energy-conservation-energy) overhead.

**Definition 9.6 (The other forms).**

[Energy](#def-g10-energy-conservation-energy) also hides in less visible costumes: *thermal energy*, the disordered agitation of a body’s molecules (the hotter, the more); *chemical energy*, stored in molecular bonds — food, wood, gasoline, batteries; *electrical energy*, carried by currents from plant to socket; *radiative energy*, carried by light even across empty space — how the Sun’s [energy](#def-g10-energy-conservation-energy) reaches us; *nuclear energy*, stored inside atomic nuclei, released in stars ([Chapter 33](https://one-course.com/books/physics/2/en/chapter/33-nuclear-energy-fission-fusion-e-mc2#ch-g12-nuclear-energy)).

## 9.2 Transfers and conversion chains

**Definition 9.7 (Transfer, conversion, chain).**

An *energy transfer* moves [energy](#def-g10-energy-conservation-energy) between systems — *reservoirs*; an *energy conversion* changes its form. A device doing either (turbine, motor, lamp, muscle) is a *converter*. A *conversion chain* pictures the journey: boxes for reservoirs and forms, labeled arrows for transfers.

![Conversion chain of a hydroelectric dam: boxes are reservoirs, arrows transfers — every real arrow leaks heat.](https://one-course.com/images/onecourse/chapters/physics-2/g10-energy-conservation/fig-da2d909e746c.svg)

*[Conversion chain](#def-g10-energy-conservation-transfer) of a hydroelectric dam: boxes are [reservoirs](#def-g10-energy-conservation-transfer), arrows transfers — every real arrow leaks heat.*

**Method 9.8 (Drawing a conversion chain).**

1. Identify the initial reservoir and form (what is consumed?).
2. Follow the [energy](#def-g10-energy-conservation-energy) [converter](#def-g10-energy-conservation-transfer) by [converter](#def-g10-energy-conservation-transfer) : one arrow per transfer, the form named in each box.
3. Add a heat arrow at every real [converter](#def-g10-energy-conservation-transfer) — some [energy](#def-g10-energy-conservation-energy) always leaks as heat ( [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) , wires, exhaust).

A pendulum runs the simplest chain of all — all potential at the top of the swing, all kinetic through the lowest point (taking $E_p = 0$ there):

![The pendulum’s budget at two positions: E_p and E_k swap while their sum (dashed) stays put.](https://one-course.com/images/onecourse/chapters/physics-2/g10-energy-conservation/fig-b41ab401ea84.svg)

![The pendulum’s budget at two positions: E_p and E_k swap while their sum (dashed) stays put.](https://one-course.com/images/onecourse/chapters/physics-2/g10-energy-conservation/fig-0679bff16adb.svg)

*The pendulum’s budget at two positions: $E_p$ and $E_k$ swap while their sum (dashed) stays put.*

## 9.3 Conservation of energy

**Theorem 9.9 (Conservation of energy).**

The total [energy](#def-g10-energy-conservation-energy) of an isolated system — one exchanging nothing with the outside — is constant. [Energy](#def-g10-energy-conservation-energy) is never created and never destroyed: it only changes form and place, and what one reservoir loses, the others gain, [joule](#def-g10-energy-conservation-energy) for [joule](#def-g10-energy-conservation-energy).

**Proof.** *Admitted at this level.* ∎

**Remark 9.10.**

No experiment has ever caught this principle failing; here we take it as given and use it numerically. [Chapter 29](https://one-course.com/books/physics/2/en/chapter/29-work-and-mechanical-energy#ch-g12-work-and-energy) derives its mechanical part ($E_k + E_p$) from Newton’s laws; the full accounting waits for the university volumes.

**Example 9.11 (Free fall, priced in joules).**

A $2.0\,\mathrm{kg}$ ball drops from $10\,\mathrm{m}$ (air resistance negligible): it starts with $E_p = 2.0 \times 9.81 \times 10 \approx 196\,\mathrm{J}$, lands with $E_k = 196\,\mathrm{J}$, so $v = \sqrt{2 \times 196/2.0} =
14\,\mathrm{m}/\mathrm{s} \approx 50\,\mathrm{km}/\mathrm{h}$. No [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force), no stopwatch: conservation alone prices the impact.

## 9.4 Efficiency

**Definition 9.12 (Efficiency).**

A [converter](#def-g10-energy-conservation-transfer) consumes $E_{\text{consumed}}$ and delivers the wanted form $E_{\text{useful}}$; its *efficiency* is

$$
\eta = \frac{E_{\text{useful}}}{E_{\text{consumed}}} < 1 .
$$

Conservation forbids $\eta > 1$, and real [converters](#def-g10-energy-conservation-transfer) never reach $1$: the missing [energy](#def-g10-energy-conservation-energy) is not destroyed — it leaves as heat.

**Example 9.13 (Where the gasoline goes).**

A [kilogram](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) of gasoline stores $45\,\mathrm{MJ}$ of [chemical energy](#def-g10-energy-conservation-forms); a car engine converts it with $\eta \approx 0.30$: $13.5\,\mathrm{MJ}$ of motion, $31.5\,\mathrm{MJ}$ heating the exhaust, the radiator and the street. Two thirds of every tank warms the atmosphere.

![The car engine’s budget, arrow widths to scale: of 45\, MJ consumed, 13.5\, MJ moves the car; the rest is heat.](https://one-course.com/images/onecourse/chapters/physics-2/g10-energy-conservation/fig-ea8d2e82b44e.svg)

*The car engine’s budget, arrow widths to scale: of $45\,\mathrm{MJ}$ consumed, $13.5\,\mathrm{MJ}$ moves the car; the rest is heat.*

**Remark 9.14 (Efficiencies multiply).**

Chained [converters](#def-g10-energy-conservation-transfer) multiply their efficiencies: a [power](#def-g10-energy-conservation-power) plant ($0.38$) feeding the grid ($0.94$) feeding a charger ($0.85$) delivers $\eta = 0.38 \times 0.94 \times 0.85 \approx 0.30$. Long chains are leaky pipes: each arrow taxes the flow.

## 9.5 Power, the watt and the kilowatt-hour

**Definition 9.15 (Power).**

*Power* is the rate at which [energy](#def-g10-energy-conservation-energy) flows, $P = E/t$, measured in *watts* ($\mathrm{W}$), with $1\,\mathrm{W} = 1\,\mathrm{J}/\mathrm{s}$. [Energy](#def-g10-energy-conservation-energy) is an amount, power a flow — a bathtub versus its tap.

**Example 9.16 (Kettle versus human).**

A $2200\,\mathrm{W}$ kettle running $150\,\mathrm{s}$ uses $E = 2200 \times 150 = 3.3 \times 10^{5}\,\mathrm{J}$. A human runs on about $10\,\mathrm{MJ}$ of food per day, an average $1.0 \times 10^{7}/86\,400 \approx 116\,\mathrm{W}$: you idle like a bright old light bulb.

**Definition 9.17 (The kilowatt-hour).**

The *kilowatt-hour* ($\mathrm{kW}\,\mathrm{h}$) is the [energy](#def-g10-energy-conservation-energy) of a $1\,\mathrm{kW}$ flow running for one hour: $1\,\mathrm{kW}\,\mathrm{h} = 1000 \times 3600\,\mathrm{J} = 3.6\,\mathrm{MJ}$ — the [unit](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) electricity meters count: [power](#def-g10-energy-conservation-power) in $\mathrm{kW}$ times hours.

**Example 9.18 (Reading the bill).**

A bill is a subtraction and a multiplication: new meter reading minus old gives the [kilowatt-hours](#def-g10-energy-conservation-kwh); times the [unit](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) price (about $0.25$ euros per $\mathrm{kW}\,\mathrm{h}$), plus a fixed subscription, gives the total. A $2\,\mathrm{kW}$ heater running $2.5\,\mathrm{h}$ nightly uses $5\,\mathrm{kW}\,\mathrm{h}$ a day: $38$ euros a month on its own.

**Example 9.19 (Orders of magnitude).**

Worth knowing by heart: lifting an apple one [metre](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit), about $1\,\mathrm{J}$; a phone battery, $10\,\mathrm{W}\,\mathrm{h} \approx 4 \times 10^{4}\,\mathrm{J}$; a cereal bowl, $1.5\,\mathrm{MJ}$; your food for a day, $10\,\mathrm{MJ}$; a [kilogram](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) of gasoline, $45\,\mathrm{MJ}$ — almost exactly a household’s daily electricity ($\approx 12\,\mathrm{kW}\,\mathrm{h}$); a lightning bolt, $5\,\mathrm{GJ}$: a hundred household-days, delivered in a flash.

![An energy ladder: each rung is a factor of 100. Everyday life spans ten orders of magnitude.](https://one-course.com/images/onecourse/chapters/physics-2/g10-energy-conservation/fig-df2e93397cd2.svg)

*An [energy](#def-g10-energy-conservation-energy) ladder: each rung is a factor of $100$. Everyday life spans ten orders of magnitude.*

## 9.6 Exercises

**Exercise 9.1 ★.**

A $1200\,\mathrm{kg}$ car drives at $90\,\mathrm{km}/\mathrm{h}$, i.e. $25\,\mathrm{m}/\mathrm{s}$: compute its [kinetic energy](#def-g10-energy-conservation-kinetic). Redo it at $130\,\mathrm{km}/\mathrm{h}$ ($36.1\,\mathrm{m}/\mathrm{s}$): what did $44\%$ more speed do to $E_k$?

**Solution of Exercise 9.1.**

$E_k = \tfrac12 \times 1200 \times 25^2 = 3.75 \times 10^{5}\,\mathrm{J}$. At $36.1\,\mathrm{m}/\mathrm{s}$: $\tfrac12 \times 1200 \times 36.1^2 \approx
7.8 \times 10^{5}\,\mathrm{J}$ — $44\%$ more speed, $(130/90)^2 \approx 2.1$ times the [kinetic energy](#def-g10-energy-conservation-kinetic): the square at work.

**Exercise 9.2 ★.**

A hiker of mass $72\,\mathrm{kg}$ climbs from altitude $1200\,\mathrm{m}$ to $1650\,\mathrm{m}$. Compute the potential [energy](#def-g10-energy-conservation-energy) gained, convert it to [kilowatt-hours](#def-g10-energy-conservation-kwh), and price it at $0.25$ euros per $\mathrm{kW}\,\mathrm{h}$.

**Solution of Exercise 9.2.**

$\Delta h = 450\,\mathrm{m}$: $E_p = 72 \times 9.81 \times 450 \approx 3.2 \times 10^{5}\,\mathrm{J}
= 3.2 \times 10^{5}/3.6 \times 10^{6} \approx 0.088\,\mathrm{kW}\,\mathrm{h}$, worth about $0.022$ euros. A day’s climbing is two cents of electricity.

**Exercise 9.3 ★.**

Convert: $1\,\mathrm{kW}\,\mathrm{h}$ to [joules](#def-g10-energy-conservation-energy); a $12\,\mathrm{W}\,\mathrm{h}$ phone battery to kilojoules; $45\,\mathrm{MJ}$ (one [kilogram](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) of gasoline) to [kilowatt-hours](#def-g10-energy-conservation-kwh); your $10\,\mathrm{MJ}$ food day to [kilowatt-hours](#def-g10-energy-conservation-kwh).

**Solution of Exercise 9.3.**

$1\,\mathrm{kW}\,\mathrm{h} = 3.6 \times 10^{6}\,\mathrm{J}$; $12\,\mathrm{W}\,\mathrm{h} = 12 \times 3600\,\mathrm{J} \approx 43\,\mathrm{kJ}$; $45\,\mathrm{MJ} = 45/3.6 = 12.5\,\mathrm{kW}\,\mathrm{h}$; $10\,\mathrm{MJ} = 10/3.6 \approx 2.8\,\mathrm{kW}\,\mathrm{h}$.

**Exercise 9.4 ★.**

A $2200\,\mathrm{W}$ kettle runs for $150\,\mathrm{s}$. Compute the [energy](#def-g10-energy-conservation-energy) used, in [joules](#def-g10-energy-conservation-energy) and in [kilowatt-hours](#def-g10-energy-conservation-kwh). Then compute the average [power](#def-g10-energy-conservation-power) of a human running on $10\,\mathrm{MJ}$ per day.

**Solution of Exercise 9.4.**

$E = 2200 \times 150 = 3.3 \times 10^{5}\,\mathrm{J}
= 3.3 \times 10^{5}/3.6 \times 10^{6} \approx 0.092\,\mathrm{kW}\,\mathrm{h}$. Human: $P = 1.0 \times 10^{7}/86\,400 \approx 116\,\mathrm{W}$.

**Exercise 9.5 ★.**

Write the [conversion chain](#def-g10-energy-conservation-transfer) ([Method 9.8](#met-g10-energy-conservation-chain)) of: a bicycle dynamo powering a lamp; a gas water heater; a solar panel charging a phone. Mark the heat leaks.

**Solution of Exercise 9.5.**

Dynamo: chemical (cyclist’s food) $\to$ kinetic (wheel) $\to$ electrical (dynamo) $\to$ radiative $+$ thermal (lamp); heat leaks at muscles, tire contact, dynamo, lamp. Gas heater: chemical (gas) $\to$ thermal (water), leak in the flue gases. Panel: radiative (Sun) $\to$ electrical (panel) $\to$ chemical (battery), heat leaks at panel and charger.

**Exercise 9.6 ★★.**

An engine burns $2.0\,\mathrm{kg}$ of gasoline ($45\,\mathrm{MJ}/\mathrm{kg}$) and delivers $27\,\mathrm{MJ}$ of motion. Compute its [efficiency](#def-g10-energy-conservation-efficiency) and the [energy](#def-g10-energy-conservation-energy) released as heat. Where does that heat go?

**Solution of Exercise 9.6.**

Consumed $2.0 \times 45 = 90\,\mathrm{MJ}$; $\eta = 27/90 = 0.30$. Heat: $90 - 27 = 63\,\mathrm{MJ}$, carried off by the exhaust gases and the radiator into the surrounding air.

**Exercise 9.7 ★★.**

A $70\,\mathrm{kg}$ cliff diver steps off a $20\,\mathrm{m}$ cliff (air resistance negligible). Using conservation, compute the [kinetic energy](#def-g10-energy-conservation-kinetic) at the water and the entry speed, in $\mathrm{m}/\mathrm{s}$ and $\mathrm{km}/\mathrm{h}$.

**Solution of Exercise 9.7.**

$E_k = E_p = 70 \times 9.81 \times 20 \approx 1.37 \times 10^{4}\,\mathrm{J}$; $v = \sqrt{2 \times 13734/70} = \sqrt{392} \approx 19.8\,\mathrm{m}/\mathrm{s}
\approx 71\,\mathrm{km}/\mathrm{h}$.

**Exercise 9.8 ★★.**

A meter read $41\,236\,\mathrm{kW}\,\mathrm{h}$ on 1 March, $41\,562\,\mathrm{kW}\,\mathrm{h}$ 30 days later: [energy](#def-g10-energy-conservation-energy) used, daily average, bill at $0.26$ euros per $\mathrm{kW}\,\mathrm{h}$ plus $12$ euros fixed, and average [power](#def-g10-energy-conservation-power) in [watts](#def-g10-energy-conservation-power)?

**Solution of Exercise 9.8.**

$41562 - 41236 = 326\,\mathrm{kW}\,\mathrm{h}$; daily $326/30 \approx
10.9\,\mathrm{kW}\,\mathrm{h}$. Bill: $326 \times 0.26 + 12 = 96.76$ euros. Average [power](#def-g10-energy-conservation-power): $326\,\mathrm{kW}\,\mathrm{h}/720\,\mathrm{h} \approx 453\,\mathrm{W}$.

**Exercise 9.9 ★★.**

An old $60\,\mathrm{W}$ bulb turns only $5\%$ of its consumption into light; an LED gives the same light from $9\,\mathrm{W}$. Compute the bulb’s light [power](#def-g10-energy-conservation-power) and the LED’s [efficiency](#def-g10-energy-conservation-efficiency); then, at $4\,\mathrm{h}$ per day, each one’s yearly consumption and the saving at $0.25$ euros per $\mathrm{kW}\,\mathrm{h}$.

**Solution of Exercise 9.9.**

*1.* Light: $0.05 \times 60 = 3.0\,\mathrm{W}$; LED: $\eta = 3/9 \approx 0.33$. *2.* Yearly ($1460\,\mathrm{h}$): bulb $60 \times 1460 = 87.6\,\mathrm{kW}\,\mathrm{h}$, LED $13.1\,\mathrm{kW}\,\mathrm{h}$; saving $74.5\,\mathrm{kW}\,\mathrm{h} \approx 19$ euros per bulb per year.

**Exercise 9.10 ★★.**

A dam passes $150\,\mathrm{m}^{3}$ of water ($1.5 \times 10^{5}\,\mathrm{kg}$) per second down a $40\,\mathrm{m}$ drop. Compute the potential [energy](#def-g10-energy-conservation-energy) released each second — the available [power](#def-g10-energy-conservation-power) — then the electrical [power](#def-g10-energy-conservation-power) at $\eta = 0.90$, and how many $525\,\mathrm{W}$-average households it supplies.

**Solution of Exercise 9.10.**

*1.* $P = 1.5 \times 10^{5} \times 9.81 \times 40 \approx
5.9 \times 10^{7}\,\mathrm{W} = 59\,\mathrm{MW}$. *2.* $0.90 \times 59 \approx 53\,\mathrm{MW}$; $5.3 \times 10^{7}/525 \approx 1.0 \times 10^{5}$ households.

**Exercise 9.11 ★★.**

A $0.20\,\mathrm{kg}$ pendulum is released $5.0\,\mathrm{cm}$ above its lowest point: compute its speed there. After a few minutes it hangs still: where did its [energy](#def-g10-energy-conservation-energy) go, and why does this not contradict [Theorem 9.9](#thm-g10-energy-conservation-conservation)?

**Solution of Exercise 9.11.**

$v = \sqrt{2 \times 9.81 \times 0.050} \approx 0.99\,\mathrm{m}/\mathrm{s}$ (the mass cancels: $\tfrac12 m v^2 = mgh$). The initial $E_p = 0.20 \times 9.81 \times 0.050 \approx 0.098\,\mathrm{J}$ ends as [thermal energy](#def-g10-energy-conservation-forms) in the air and the pivot: the pendulum is not isolated, and the total — swing plus heat — is exactly conserved.

**Exercise 9.12 ★★★.**

A coal plant ($\eta_1 = 0.38$), the grid ($\eta_2 = 0.94$) and a charger ($\eta_3 = 0.85$) recharge a $12\,\mathrm{W}\,\mathrm{h}$ phone battery. Compute the overall [efficiency](#def-g10-energy-conservation-efficiency); the [chemical energy](#def-g10-energy-conservation-forms) consumed at the plant, in $\mathrm{W}\,\mathrm{h}$ and kilojoules; and the coal ($25\,\mathrm{MJ}/\mathrm{kg}$) per charge, then per year of daily charges.

**Solution of Exercise 9.12.**

*1.* $\eta = 0.38 \times 0.94 \times 0.85 \approx 0.304$; consumed $12/0.304 \approx 39.5\,\mathrm{W}\,\mathrm{h}
= 39.5 \times 3600 \approx 142\,\mathrm{kJ}$. *2.* $1.42 \times 10^{5}/2.5 \times 10^{7} \approx 5.7\,\mathrm{g}$ of coal per charge; $\times 365 \approx 2.1\,\mathrm{kg}$ per year.

**Exercise 9.13 ★★★.**

A lightning bolt carries about $5\,\mathrm{GJ}$. Convert to [kilowatt-hours](#def-g10-energy-conservation-kwh) and to days of a $12.6\,\mathrm{kW}\,\mathrm{h}$-per-day household. It is delivered in about $100\,\text{µ}\mathrm{s}$: compute the [power](#def-g10-energy-conservation-power), compare it with humanity’s average electric [power](#def-g10-energy-conservation-power) ($3\,\mathrm{TW}$), and explain why nobody harvests lightning.

**Solution of Exercise 9.13.**

*1.* $5 \times 10^{9}/3.6 \times 10^{6} \approx 1400\,\mathrm{kW}\,\mathrm{h}$; $1400/12.6 \approx 110$ days. *2.* $P = 5 \times 10^{9}/1 \times 10^{-4} = 5 \times 10^{13}\,\mathrm{W} = 50\,\mathrm{TW}$ — seventeen times humanity’s average electric [power](#def-g10-energy-conservation-power), for $100\,\text{µ}\mathrm{s}$, at an unpredictable place. Lightning is much [power](#def-g10-energy-conservation-power) but little [energy](#def-g10-energy-conservation-energy), delivered uselessly fast: no [converter](#def-g10-energy-conservation-transfer) accepts such a spike.

**Exercise 9.14 ★★★.**

A $1300\,\mathrm{kg}$ car brakes from $110\,\mathrm{km}/\mathrm{h}$ ($30.6\,\mathrm{m}/\mathrm{s}$) to rest. Compute the [kinetic energy](#def-g10-energy-conservation-kinetic) dissipated; what form does it take, and where? An electric car recovers $60\%$ into its battery: [energy](#def-g10-energy-conservation-energy) per stop, and stops needed to refill one [kilowatt-hour](#def-g10-energy-conservation-kwh)?

**Solution of Exercise 9.14.**

*1.* $E_k = \tfrac12 \times 1300 \times 30.6^2 \approx
6.1 \times 10^{5}\,\mathrm{J}$, converted to [thermal energy](#def-g10-energy-conservation-forms) in the brake discs and pads (then the air). *2.* $0.60 \times 6.1 \times 10^{5}\,\mathrm{J} \approx 3.7 \times 10^{5}\,\mathrm{J}$ per stop; $3.6 \times 10^{6}/3.7 \times 10^{5} \approx 10$ stops per [kilowatt-hour](#def-g10-energy-conservation-kwh).

**Exercise 9.15 ★★★.**

A cyclist and bicycle ($85\,\mathrm{kg}$ in all) climb a mountain pass, gaining $1200\,\mathrm{m}$ of altitude; muscles convert food with $\eta \approx 0.25$. Compute the mechanical [energy](#def-g10-energy-conservation-energy) of the climb, the food [energy](#def-g10-energy-conservation-energy) it demands, the equivalent in $1.5\,\mathrm{MJ}$ cereal bowls, and the gasoline mass ($45\,\mathrm{MJ}/\mathrm{kg}$) storing the same [energy](#def-g10-energy-conservation-energy). Comment.

**Solution of Exercise 9.15.**

$E_p = 85 \times 9.81 \times 1200 \approx 1.0\,\mathrm{MJ}$; food: $1.0/0.25 = 4.0\,\mathrm{MJ}$, i.e. $4.0/1.5 \approx 2.7$ cereal bowls — or $4.0/45 \approx 0.09\,\mathrm{kg}$ of gasoline. A whole morning of honest effort fits in a coffee cup of fuel: [chemical energy](#def-g10-energy-conservation-forms) is absurdly concentrated.

## 9.7 Problem: Powering a house for a day

**Problem 9.1.**

Weekend problem — from cereal bowl to rooftop panels: a full energy audit of one ordinary day at home, and what a joule costs depending on who sells it

One ordinary Tuesday, a family sets out to find where its electricity goes, armed with this chapter, the hallway meter and the back of the bill. Tonight’s verdict: their day equals one [kilogram](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) of gasoline, sixteen square [metres](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) of sunshine — or five cyclists pedaling around the clock.

**Part I — The audit.** The day’s inventory: fridge $100\,\mathrm{W}$ for $24\,\mathrm{h}$; water heater $2400\,\mathrm{W}$ for $2.5\,\mathrm{h}$; oven $2000\,\mathrm{W}$ for $1.0\,\mathrm{h}$; washing machine $500\,\mathrm{W}$ for $2.0\,\mathrm{h}$; lighting $60\,\mathrm{W}$ for $5.0\,\mathrm{h}$; screens $150\,\mathrm{W}$ for $6.0\,\mathrm{h}$.

1. Compute each appliance’s [energy](#def-g10-energy-conservation-energy) for the day, in $\mathrm{kW}\,\mathrm{h}$ .
2. Total the day in [kilowatt-hours](#def-g10-energy-conservation-kwh) , then in megajoules; compare with one [kilogram](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) of gasoline ( $45\,\mathrm{MJ}$ ).
3. Compute the house’s average [power](#def-g10-energy-conservation-power) over the $24\,\mathrm{h}$ ; what fraction of a running kettle is that?
4. Which single appliance dominates? Recompute the daily total if the heater ran only $1.25\,\mathrm{h}$ .
5. Standby lights draw a permanent $15\,\mathrm{W}$ : daily [energy](#def-g10-energy-conservation-energy) , share of the total, yearly cost at $0.25$ euros per $\mathrm{kW}\,\mathrm{h}$ ?

**Part II — The bill.**

6. In 30 days the meter went from $56\,214\,\mathrm{kW}\,\mathrm{h}$ to $56\,604\,\mathrm{kW}\,\mathrm{h}$ : consumption, and daily average checked against the audit?
7. Compute the bill: $0.25$ euros per $\mathrm{kW}\,\mathrm{h}$ plus a fixed $14$ euros subscription.
8. Price of a megajoule from the grid? From gasoline, at $1.80$ euros per litre storing $34\,\mathrm{MJ}$ ?
9. A $1.5\,\mathrm{MJ}$ cereal bowl costs about $0.50$ euros: its megajoule price? Rank the three sellers of [joules](#def-g10-energy-conservation-energy) .

**Part III — The rooftop.**

10. Full sunlight delivers about $1000\,\mathrm{W}$ per square [metre](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) ; panels convert it with $\eta = 0.20$ : electrical [power](#def-g10-energy-conservation-power) of $1\,\mathrm{m}^{2}$ in full sun?
11. The region averages the equivalent of $4.0\,\mathrm{h}$ of full sun per day: daily yield of $1\,\mathrm{m}^{2}$ ?
12. What panel area covers the family’s $12.6\,\mathrm{kW}\,\mathrm{h}$ day? The south roof offers $20\,\mathrm{m}^{2}$ : does it fit?
13. In December the equivalent drops to $1.5\,\mathrm{h}$ : the panels’ production, its share of the need, and what fills the gap?
14. Manufacturing the panels cost about $700\,\mathrm{kW}\,\mathrm{h}$ per square [metre](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) : [energy](#def-g10-energy-conservation-energy) payback time, using question 11?

**Part IV — The cereal bowl.**

15. A generator bicycle yields $100\,\mathrm{W}$ of electricity: how many pedaling hours make $12.6\,\mathrm{kW}\,\mathrm{h}$ — how many people pedaling $24\,\mathrm{h}$ non-stop?
16. Muscles run at $\eta \approx 0.25$ : the food [energy](#def-g10-energy-conservation-energy) behind those $12.6\,\mathrm{kW}\,\mathrm{h}$ , in megajoules and in cereal bowls?
17. Cost of the pedaled day (bowls at $0.50$ euros), against what the grid charges for the same day?
18. A $5\,\mathrm{GJ}$ lightning bolt equals how many house-days? Given its $100\,\text{µ}\mathrm{s}$ duration, why can it still not [power](#def-g10-energy-conservation-power) the house?
19. Joules are never lost, yet grid, gasoline and food [joules](#def-g10-energy-conservation-energy) sell at very different prices: what do you actually pay for?
20. Finale: state the day’s verdict in one sentence — one [kilogram](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) of gasoline, sixteen square [metres](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) of sunny roof or five round-the-clock cyclists — and name the two quantities (one stock, one flow) never to confuse again.

**Solution of Problem 9.1.**

**1.** Fridge $0.1 \times 24 = 2.4\,\mathrm{kW}\,\mathrm{h}$; heater $2.4 \times 2.5 = 6.0\,\mathrm{kW}\,\mathrm{h}$; oven $2.0\,\mathrm{kW}\,\mathrm{h}$; washing machine $1.0\,\mathrm{kW}\,\mathrm{h}$; lighting $0.3\,\mathrm{kW}\,\mathrm{h}$; screens $0.9\,\mathrm{kW}\,\mathrm{h}$. **2.** Total $12.6\,\mathrm{kW}\,\mathrm{h}$ $= 12.6 \times 3.6 \approx
45\,\mathrm{MJ}$: one [kilogram](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) of gasoline, to the [joule](#def-g10-energy-conservation-energy). **3.** $12\,600\,\mathrm{W}\,\mathrm{h}/24\,\mathrm{h} = 525\,\mathrm{W}$ — the house idles at a quarter of a kettle. **4.** The water heater ($6.0\,\mathrm{kW}\,\mathrm{h}$, nearly half). At $1.25\,\mathrm{h}$: heater $3.0\,\mathrm{kW}\,\mathrm{h}$, total $9.6\,\mathrm{kW}\,\mathrm{h}$. **5.** $15 \times 24 = 0.36\,\mathrm{kW}\,\mathrm{h}$ per day, about $2.8\%$ of the metered total; yearly $0.36 \times 365 \approx
131\,\mathrm{kW}\,\mathrm{h} \approx 33$ euros — for lights nobody watches. **6.** $56604 - 56214 = 390\,\mathrm{kW}\,\mathrm{h}$; $390/30 = 13.0\,\mathrm{kW}\,\mathrm{h}$ per day: the audit’s $12.6\,\mathrm{kW}\,\mathrm{h}$ plus the $0.36\,\mathrm{kW}\,\mathrm{h}$ of standby — confirmed. **7.** $390 \times 0.25 + 14 = 111.50$ euros. **8.** Grid: $0.25/3.6 \approx 0.07$ euros per megajoule. Gasoline: $1.80/34 \approx 0.05$ euros per megajoule. **9.** Bowl: $0.50/1.5 \approx 0.33$ euros per megajoule — five times the grid. Cheapest first: gasoline, grid, then food, far behind. **10.** $0.20 \times 1000 = 200\,\mathrm{W}$ per square [metre](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit). **11.** $200 \times 4.0 = 800\,\mathrm{W}\,\mathrm{h} = 0.80\,\mathrm{kW}\,\mathrm{h}$ per square [metre](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) per day. **12.** $12.6/0.80 = 15.75 \approx 16\,\mathrm{m}^{2}$: it fits the $20\,\mathrm{m}^{2}$ roof. **13.** $16 \times 0.2 \times 1.5 = 4.8\,\mathrm{kW}\,\mathrm{h}$: about $38\%$ of the need; the grid (or a battery charged in better months) must supply the rest. **14.** One square [metre](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) yields $0.80 \times 365 \approx
292\,\mathrm{kW}\,\mathrm{h}$ per year; payback $700/292 \approx 2.4$ years — short against a 25-year panel lifetime. **15.** $12.6\,\mathrm{kW}\,\mathrm{h}/0.1\,\mathrm{kW} = 126\,\mathrm{h}$: more than five people ($126/24 = 5.25$) pedaling day and night. **16.** Food $= 12.6/0.25 = 50.4\,\mathrm{kW}\,\mathrm{h} \approx
181\,\mathrm{MJ}$, i.e. $181/1.5 \approx 121$ cereal bowls. **17.** $121 \times 0.50 \approx 60$ euros, against $12.6 \times 0.25 \approx 3.15$ euros from the grid: pedal [power](#def-g10-energy-conservation-power) costs about $19$ times more — before paying the cyclists. **18.** $5000/45 \approx 110$ house-days. But $5 \times 10^{9}/1 \times 10^{-4} = 5 \times 10^{13}\,\mathrm{W}$ for $100\,\text{µ}\mathrm{s}$, at a random place: no [converter](#def-g10-energy-conservation-transfer) or battery can drink from that hose. A house needs a steady $525\,\mathrm{W}$ flow, not a spike. **19.** Not [joules](#def-g10-energy-conservation-energy) — they are conserved and identical. You pay for their *form and availability*: [energy](#def-g10-energy-conservation-energy) that is concentrated, storable and on demand costs the whole [conversion chain](#def-g10-energy-conservation-transfer) that made it so. **20.** One family day $= 12.6\,\mathrm{kW}\,\mathrm{h} \approx
45\,\mathrm{MJ}$: one [kilogram](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) of gasoline, sixteen square [metres](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) of sunny roof, or five round-the-clock cyclists. The stock is [energy](#def-g10-energy-conservation-energy) ([joules](#def-g10-energy-conservation-energy), [kilowatt-hours](#def-g10-energy-conservation-kwh)); the flow is [power](#def-g10-energy-conservation-power) ([watts](#def-g10-energy-conservation-power)) — the bathtub and the tap, never again confused.