---
title: "Forms of Energy and Conversions"
book: "Primary & Middle School Physics"
subject: physics
language: en
chapter: 70
exercises: 12
source: https://one-course.com/books/physics/1/en/chapter/70-forms-of-energy-and-conversions
---

# Chapter 70 — Forms of Energy and Conversions

[Energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) has run through this book like a river under a city — glimpsed at every corner, never yet mapped whole. The time has come. Every form gets its name, one of them gets this year’s second great formula, and over them all presides the mightiest bookkeeping law in science: the total never changes. Ever.

## 70.1 The forms, assembled

**Definition 70.1 (The forms of energy).**

The forms of [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy), formally presented: *[kinetic energy](https://one-course.com/books/physics/1/en/chapter/66-kinetic-energy-and-road-safety#prop-g9-kinetic-energy-safety-formula)* — motion’s share, $\frac12 m v^2$; *potential energy* — stored by position or arrangement (the lifted, the stretched, the [compressed](https://one-course.com/books/physics/1/en/chapter/45-states-of-matter-and-changes-of-state#prop-g7-states-of-matter-squeeze)); *thermal [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy)* — the disordered molecular jiggling that [temperature](https://one-course.com/books/physics/1/en/chapter/15-temperature-and-thermometers#def-g3-temperature-thermometers-temperature) [measures](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring); *chemical [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy)* — stored in matter’s arrangements: food, fuels, batteries; *electrical [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy)* — the marching current’s deliverable; *radiant [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy)* — carried by [light](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source) itself, sunshine’s shipping form; and *nuclear [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy)* — locked in the atom’s core, the deepest store humans have opened.

**Proposition 70.2 (Gravitational potential energy).**

Lifting a [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) $m$ (in $\mathrm{kg}$) through a height $h$ (in $\mathrm{m}$) against gravity’s $g$ stores in it the [potential energy](#def-g9-energy-conversions-forms)

$$
E_p = m \times g \times h
$$

[joules](https://one-course.com/books/physics/1/en/chapter/66-kinetic-energy-and-road-safety#def-g9-kinetic-energy-safety-joule) — recoverable in full on the way back down. The lifted axe, the drawn-back swing, the mountain lake: all hold $m g h$ on account. (This is the borrowed formula of the road-safety posters — now officially yours.)

**Example 70.3 (The great trade: EpE_pEp​ against EkE_kEk​).**

Drop a ball from height $h$: as it falls, its account transfers, [joule](https://one-course.com/books/physics/1/en/chapter/66-kinetic-energy-and-road-safety#def-g9-kinetic-energy-safety-joule) by [joule](https://one-course.com/books/physics/1/en/chapter/66-kinetic-energy-and-road-safety#def-g9-kinetic-energy-safety-joule), from potential to kinetic. At the ground, the whole $m g h$ has become $\frac12 m v^2$ — set them equal and the [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) cancels:

$$
v = \sqrt{2 g h},
$$

this year’s square root earning its keep. From $20\,\mathrm{m}$: $v = \sqrt{2 \times 9.8 \times 20} \approx 20\,\mathrm{m}/\mathrm{s}$ — seventy kilometres per hour, [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) be what it may. The pendulum plays the trade both ways forever (minus a whisper to the [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air)); the skateboarder in the half-pipe likewise: height to [speed](https://one-course.com/books/physics/1/en/chapter/52-motion-graphs-and-average-speed#def-g7-motion-average-speed-formula) to height, the two formulas passing one sum back and forth.

![The rollercoaster’s ledger: height spent buys speed, speed spent buys height back — one sum, two accounts, the total riding unchanged (but for friction’s quiet skim).](https://one-course.com/images/onecourse/chapters/physics-1/g9-energy-conversions/fig-e45a67fd4e1a.svg)

*The rollercoaster’s ledger: height spent buys [speed](https://one-course.com/books/physics/1/en/chapter/52-motion-graphs-and-average-speed#def-g7-motion-average-speed-formula), [speed](https://one-course.com/books/physics/1/en/chapter/52-motion-graphs-and-average-speed#def-g7-motion-average-speed-formula) spent buys height back — one sum, two accounts, the total riding unchanged (but for friction’s quiet skim).*

## 70.2 The great law

**Proposition 70.4 (Conservation of energy).**

In every process ever examined — collisions and chemistry, machines and storms, [stars](https://one-course.com/books/physics/1/en/chapter/26-the-solar-system#def-g4-solar-system-star) and cells — [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) changes form and owner, but the *total* is exactly conserved: none created, none destroyed, every [joule](https://one-course.com/books/physics/1/en/chapter/66-kinetic-energy-and-road-safety#def-g9-kinetic-energy-safety-joule) accounted for. This is the deepest bookkeeping law in physics: it retired the forever-machines of the childhood chapters, it audits every chain of this book, and no experiment in three centuries has caught it in an error. (Why nature keeps this law so faithfully is a profound question with a beautiful answer — one of the jewels of the university years.)

**Method 70.5 (Auditing any process).**

The physicist’s universal bookkeeping:

1. fence off the system — what is inside the audit;
2. list the [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) accounts before: each form, each store;
3. list them after — including, always, the thermal small change: friction’s warmth, [sound](https://one-course.com/books/physics/1/en/chapter/19-sound-around-us#def-g3-sound-around-us-sound) ’s whisper;
4. balance the books: the totals must match. A deficit means a form went uncounted — look again for the leak; it is usually warm.

**Example 70.6 (Audits, worked).**

The braking car: kinetic account emptied, thermal account at the discs credited in full — retirement, not destruction. The bouncing ball that dies away: each bounce skims kinetic into warmth of ball and floor; the childhood answer, now auditable. Your breakfast cycling uphill: chemical store down, potential account up ($m g h$ of you and bicycle), thermal share radiated — muscles run warm. Every mystery of the early [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) chapters closes under the same three columns.

## 70.3 Efficiency

**Definition 70.7 (Efficiency).**

The *efficiency* of a converter is the fraction of its input [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) delivered in the wanted form:

$$
\text{efficiency} =
\frac{\text{useful energy out}}{\text{total energy in}},
$$

a number between $0$ and $1$ (or its percentage). The remainder is not destroyed — conservation forbids — but escapes in unwanted forms, nearly always warmth.

**Example 70.8 (Report cards).**

The glowing ancestor [bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb): five percent [light](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source), ninety-five warmth — a heater with a hobby. Its modern successor: thirty-plus percent, the rest still warmth. A car engine: about a third useful motion, two thirds out the radiator and exhaust. An electric [motor](https://one-course.com/books/physics/1/en/chapter/41-electric-circuits-and-safety#ex-g6-circuits-and-safety-motor): above ninety. The great [alternators](https://one-course.com/books/physics/1/en/chapter/69-from-power-plant-to-home-the-alternator#def-g9-alternator-alternator): near ninety-nine — civilization’s finest converters. No honest converter reaches one hundred: some leak to warmth is nature’s universal commission.

![An efficiency diagram: one hundred joules in, thirty-five delivered as wanted, sixty-five leaked as warmth — and not one joule missing from the sum.](https://one-course.com/images/onecourse/chapters/physics-1/g9-energy-conversions/fig-11ae631f4cc3.svg)

*An [efficiency](#def-g9-energy-conversions-efficiency) diagram: one hundred [joules](https://one-course.com/books/physics/1/en/chapter/66-kinetic-energy-and-road-safety#def-g9-kinetic-energy-safety-joule) in, thirty-five delivered as wanted, sixty-five leaked as warmth — and not one [joule](https://one-course.com/books/physics/1/en/chapter/66-kinetic-energy-and-road-safety#def-g9-kinetic-energy-safety-joule) missing from the sum.*

![Two converters in one field: sunlight to electrical energy in the panels, moving air to electrical energy in the turbines.](https://one-course.com/images/onecourse/chapters/physics-1/g9-energy-conversions/img-b468457089de.jpg)

*Two converters in one field: sunlight to electrical [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) in the panels, moving [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air) to electrical [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) in the turbines.*

**Remark 70.9 (What “saving energy” really means).**

If conservation guards every [joule](https://one-course.com/books/physics/1/en/chapter/66-kinetic-energy-and-road-safety#def-g9-kinetic-energy-safety-joule), why must anyone “save [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy)”? Because the law preserves *amount*, not *usefulness*. Concentrated stores — fuel, charge, height — can be spent into warmth spread thinly through the world, and spread warmth, though every [joule](https://one-course.com/books/physics/1/en/chapter/66-kinetic-energy-and-road-safety#def-g9-kinetic-energy-safety-joule) of it survives, can no longer drive kettles or trains: the river ran downhill. Civilization’s [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) problem is not a shortage of [joules](https://one-course.com/books/physics/1/en/chapter/66-kinetic-energy-and-road-safety#def-g9-kinetic-energy-safety-joule) — the Sun showers us with thousands of times our needs — but the husbanding of *useful* ones. That idea, made precise, is one of the most celebrated stories in physics; it waits in the university years under its famous name.

## 70.4 Exercises

**Exercise 70.1 ★.**

Name the seven forms of the catalog, each with one household example.

**Solution of Exercise 70.1.**

Kinetic (the rolling ball), potential (the raised bucket), thermal (the warm bath), chemical (breakfast, the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery)), electrical (the wall’s supply), radiant (sunshine through the window), nuclear (the power station’s core).

**Exercise 70.2 ★.**

Compute $E_p$: a $2\,\mathrm{kg}$ bucket raised $10\,\mathrm{m}$; a $60\,\mathrm{kg}$ hiker atop a $500\,\mathrm{m}$ climb.

**Solution of Exercise 70.2.**

Bucket: $2 \times 9.8 \times 10 = 196\,\mathrm{J}$. Hiker: $60
\times 9.8 \times 500 = 2.9 \times 10^{5}\,\mathrm{J}$ — nearly three hundred kilojoules of altitude.

**Exercise 70.3 ★.**

State the conservation law, and what it did to the forever-machines of your childhood chapters.

**Solution of Exercise 70.3.**

In every process, [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) changes form and owner but the total is exactly conserved. The forever-machines died of it: a machine that outputs more than it stores or receives would create [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) — forbidden without appeal.

**Exercise 70.4 ★.**

Audit the pendulum through one swing: accounts at the top, at the bottom, and the slow overall drift — to where?

**Solution of Exercise 70.4.**

Top of swing: all potential, momentarily still. Bottom: all kinetic, fastest. The drift: each pass skims a little into warmth of [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air) and [pivot](https://one-course.com/books/physics/1/en/chapter/17-levers-and-balance-scales#def-g3-levers-and-scales-lever), and the swing dies down — accounts transferred, total intact.

**Exercise 70.5 ★.**

Use $v = \sqrt{2 g h}$: the splash [speed](https://one-course.com/books/physics/1/en/chapter/52-motion-graphs-and-average-speed#def-g7-motion-average-speed-formula) from a $5\,\mathrm{m}$ diving board — and why the diver’s [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) never entered.

**Solution of Exercise 70.5.**

$v = \sqrt{2 \times 9.8 \times 5} \approx 9.9\,\mathrm{m}/\mathrm{s}$ — about $36\,\mathrm{km}/\mathrm{h}$. The [mass](https://one-course.com/books/physics/1/en/chapter/20-measuring-mass#def-g3-measuring-mass-mass) canceled when $m g h$ was set against $\frac12 m v^2$: heavy and [light](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source) divers splash at the same [speed](https://one-course.com/books/physics/1/en/chapter/52-motion-graphs-and-average-speed#def-g7-motion-average-speed-formula).

**Exercise 70.6 ★.**

Define [efficiency](#def-g9-energy-conversions-efficiency). A [motor](https://one-course.com/books/physics/1/en/chapter/41-electric-circuits-and-safety#ex-g6-circuits-and-safety-motor) takes $200\,\mathrm{J}$ and delivers $170\,\mathrm{J}$ of motion: [efficiency](#def-g9-energy-conversions-efficiency), and the fate of the rest?

**Solution of Exercise 70.6.**

Useful over total: $170 \div 200 = 0.85$ — eighty-five percent. The missing $30\,\mathrm{J}$: warmth in windings and bearings.

**Exercise 70.7 ★.**

Why was the ancestor [bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) fairly called “a heater with a hobby”? Give both percentages.

**Solution of Exercise 70.7.**

Five percent of its input left as [light](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source); ninety-five left as warmth. A device sorted by its outputs was a heater — that happened to glow.

**Exercise 70.8 ★.**

Write the full conversion chain of the cyclist’s climb — breakfast to summit — naming each form and each leak.

**Solution of Exercise 70.8.**

Chemical (breakfast) $\to$ muscles’ work $\to$ [kinetic energy](https://one-course.com/books/physics/1/en/chapter/66-kinetic-energy-and-road-safety#prop-g9-kinetic-energy-safety-formula) of rider and machine $\to$ stored as [potential energy](#def-g9-energy-conversions-forms), $m g
h$, at the summit — with thermal leaks throughout: warm muscles, warm [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air) stirred, warm tires. Total: conserved.

**Exercise 70.9 ★★.**

The dam’s ledger: $1000\,\mathrm{kg}$ of water falls $50\,\mathrm{m}$ through the turbines. Compute the delivered $E_p$; at ninety-percent conversion, the electrical [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) per tonne — and the tonnes per second needed for a $4.4 \times 10^{6}\,\mathrm{W}$ output.

**Solution of Exercise 70.9.**

Per tonne: $1000 \times 9.8 \times 50 = 4.9 \times 10^{5}\,\mathrm{J}$; at ninety percent, about $4.4 \times 10^{5}\,\mathrm{J}$ electrical. For $4.4 \times 10^{6}\,\mathrm{W}$: $4.4 \times 10^{6} \div 4.4 \times 10^{5} = 10$ tonnes per second through the turbines.

**Exercise 70.10 ★★.**

A $0.5\,\mathrm{kg}$ ball dropped from $2\,\mathrm{m}$ rebounds to $1.2\,\mathrm{m}$. Audit the bounce: energies before and after, the skim’s size and destination — and the rebound height after which no audit will find kinetic or [potential energy](#def-g9-energy-conversions-forms) left.

**Solution of Exercise 70.10.**

Before: $0.5 \times 9.8 \times 2 = 9.8\,\mathrm{J}$. After the bounce: $0.5 \times 9.8 \times 1.2 \approx 5.9\,\mathrm{J}$. The skim: about $3.9\,\mathrm{J}$ into warmth of ball and floor (and a click of [sound](https://one-course.com/books/physics/1/en/chapter/19-sound-around-us#def-g3-sound-around-us-sound) — itself soon warmth). The audit ends when rebounds reach zero height: every [joule](https://one-course.com/books/physics/1/en/chapter/66-kinetic-energy-and-road-safety#def-g9-kinetic-energy-safety-joule) of the original $9.8$ then rests in the thermal account.

**Exercise 70.11 ★★.**

“The lift converts electrical [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) into [potential energy](#def-g9-energy-conversions-forms) at ninety percent [efficiency](#def-g9-energy-conversions-efficiency).” Compute the electrical [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) to raise a $500\,\mathrm{kg}$ cabin $30\,\mathrm{m}$ — and reconcile the extra ten percent with the conservation law.

**Solution of Exercise 70.11.**

$E_p = 500 \times 9.8 \times 30 = 1.47 \times 10^{5}\,\mathrm{J}$; at ninety percent the meter must supply $1.47 \times 10^{5} \div 0.9 \approx
1.6 \times 10^{5}\,\mathrm{J}$. The extra tenth is not lost to the universe — it warms [motor](https://one-course.com/books/physics/1/en/chapter/41-electric-circuits-and-safety#ex-g6-circuits-and-safety-motor), gears and shaft: conservation holds; the lift’s ledger simply includes a thermal line.

**Exercise 70.12 ★★★.**

The half-pipe skater starts from rest at the left rim, height $4\,\mathrm{m}$. Predict the crossing [speed](https://one-course.com/books/physics/1/en/chapter/52-motion-graphs-and-average-speed#def-g7-motion-average-speed-formula) at the bottom, the height reached on the right — ideally and really — and sketch the long-term fate of the motion and of its [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy). Then explain, with [Remark 70.9](#rem-g9-energy-conversions-saving), why no push at the rim can ever be recovered in full: what has the world’s ledger gained, and what has the skater irrecoverably lost?

**Solution of Exercise 70.12.**

Bottom [speed](https://one-course.com/books/physics/1/en/chapter/52-motion-graphs-and-average-speed#def-g7-motion-average-speed-formula): $v = \sqrt{2 \times 9.8 \times 4} \approx
8.9\,\mathrm{m}/\mathrm{s}$. Ideally the right rim is regained at exactly $4\,\mathrm{m}$; really a little lower, each crossing lower still, until the skater rocks to rest at the bottom — the whole $m g h$ retired into warmth of wheels, ramp and [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air). Nothing was destroyed, but everything useful was spent: the world’s ledger gained spread-out warmth, and the skater lost the concentrated, reusable height — the amount survives; the usefulness does not return.

## 70.5 Problem: The Theme Park Commission

**Problem 70.1.**

Weekend problem — the theme park hires a physics consultant; coasters, towers and the ledger that cannot lie

The park’s new attractions must pass your [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) audit before opening day. Take $g = 9.8\,\mathrm{N}/\mathrm{kg}$; ignore friction until it refuses to be ignored.

**Part I — The Great Drop.** A $2000\,\mathrm{kg}$ coaster train is winched to a $45\,\mathrm{m}$ summit and released from rest.

1. The train’s $E_p$ at the summit?
2. Its ideal [speed](https://one-course.com/books/physics/1/en/chapter/52-motion-graphs-and-average-speed#def-g7-motion-average-speed-formula) at the bottom of the drop, by the mass-canceling formula — in $\mathrm{m}/\mathrm{s}$ and $\mathrm{km}/\mathrm{h}$ ?
3. The designers add a second hill of $50\,\mathrm{m}$ just after the drop. Veto it with the ledger — and state the tallest second hill the ideal train could crest.
4. The measured bottom [speed](https://one-course.com/books/physics/1/en/chapter/52-motion-graphs-and-average-speed#def-g7-motion-average-speed-formula) is $28\,\mathrm{m}/\mathrm{s}$ , not the ideal. Compute both kinetic energies and charge the difference to the proper account.

**Part II — The winch and the bill.**

5. The winch lifts the train to its summit in $60\,\mathrm{s}$ . At perfect [efficiency](#def-g9-energy-conversions-efficiency) , its [power](https://one-course.com/books/physics/1/en/chapter/68-electric-power-and-energy#def-g9-electric-power-energy-power) ?
6. The real winch draws $20\,\mathrm{kW}$ . Its [efficiency](#def-g9-energy-conversions-efficiency) ?
7. Each ride runs every four minutes, ten [hours](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#ex-g2-measuring-time-clock) a day. The winch’s daily [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) in [kilowatt-hours](https://one-course.com/books/physics/1/en/chapter/68-electric-power-and-energy#def-g9-electric-power-energy-kwh) (real winch, count the lifts), and its cost at $0.25$ per [unit](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) ?
8. A designer proposes recovering the train’s arrival [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) with a generator-brake feeding the grid — “the ride pays for itself!” Audit the dream: what fraction could realistically return, and which law forbids the full circle?

**Part III — The freefall tower and the report.** The $60\,\mathrm{m}$ tower drops a $500\,\mathrm{kg}$ gondola into magnetic brakes at $10\,\mathrm{m}$.

9. [Speed](https://one-course.com/books/physics/1/en/chapter/52-motion-graphs-and-average-speed#def-g7-motion-average-speed-formula) entering the brake zone (a $50\,\mathrm{m}$ ideal fall)?
10. [Energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) the brakes must retire, and where induction sends it. (The brakes are the [alternator](https://one-course.com/books/physics/1/en/chapter/69-from-power-plant-to-home-the-alternator#def-g9-alternator-alternator) chapter’s cousins — eddies of induced current heating metal.)
11. Riders scream that they were “weightless” during the fall. From the falling-companions idea of the [gravitation](https://one-course.com/books/physics/1/en/chapter/63-gravitation#def-g9-gravitation-gravitation) chapter, adjudicate the claim in one sentence.
12. Sign the commission’s report: three sentences — the trade the coaster lives by, the law every attraction obeyed, and the account that quietly taxed them all.

**Solution of Problem 70.1.**

**1.** $E_p = 2000 \times 9.8 \times 45 =
8.8 \times 10^{5}\,\mathrm{J}$. **2.** $v = \sqrt{2 \times 9.8 \times 45} \approx
29.7\,\mathrm{m}/\mathrm{s} \approx 107\,\mathrm{km}/\mathrm{h}$. **3.** A $50\,\mathrm{m}$ crest demands more [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) than the summit granted: the ledger forbids it. From a $45\,\mathrm{m}$ start at rest, no later hill may exceed $45\,\mathrm{m}$ — and really, friction demands visibly less. **4.** Measured: $0.5 \times 2000 \times 28^2 =
7.8 \times 10^{5}\,\mathrm{J}$ against the ideal $8.8 \times 10^{5}\,\mathrm{J}$: about $1 \times 10^{5}\,\mathrm{J}$ charged to the thermal account — rails, wheels and [air](https://one-course.com/books/physics/1/en/chapter/10-air-around-us#def-g2-air-around-us-air), warmed. **5.** $P = 8.8 \times 10^{5} \div 60 \approx
1.5 \times 10^{4}\,\mathrm{W}$ — some fifteen kilowatts. **6.** $14.7 \div 20 \approx 0.74$: seventy-four percent. **7.** Ten [hours](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#ex-g2-measuring-time-clock) at one lift per four minutes: $150$ lifts; each costs $20 \times \tfrac{1}{60} =
0.33\,\mathrm{kWh}$; daily $150 \times 0.33 = 50\,\mathrm{kWh}$ — $12.5$ currency [units](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring). **8.** A generous brake-generator might recover a fair share of the arrival [kinetic energy](https://one-course.com/books/physics/1/en/chapter/66-kinetic-energy-and-road-safety#prop-g9-kinetic-energy-safety-formula) — but every stage (rails, brake, generator, transformer) pays its thermal commission, and the recovered [joules](https://one-course.com/books/physics/1/en/chapter/66-kinetic-energy-and-road-safety#def-g9-kinetic-energy-safety-joule) can never equal the winch’s spending: conservation allows recycling; efficiency-less-than-one forbids the full circle. The ride can defray itself, never pay for itself. **9.** $v = \sqrt{2 \times 9.8 \times 50} \approx
31.3\,\mathrm{m}/\mathrm{s}$. **10.** Kinetic at entry $0.5 \times 500 \times 980 =
2.45 \times 10^{5}\,\mathrm{J}$, plus the last ten [metres](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units)’ $500 \times 9.8
\times 10 = 4.9 \times 10^{4}\,\mathrm{J}$: about $2.9 \times 10^{5}\,\mathrm{J}$ for the brakes — sent, by induced eddy currents, into warmth of the brake fins: induction as retirement plan. **11.** The claim stands, felt if not worded exactly: gondola and riders fell together, and falling companions press on nothing — the scream is shared free fall, gravity present in full and unfelt. **12.** For example: “Our coasters live by one trade: height for [speed](https://one-course.com/books/physics/1/en/chapter/52-motion-graphs-and-average-speed#def-g7-motion-average-speed-formula) and back, $m g h$ against $\frac12 m v^2$. Every attraction obeyed one law: the [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) total, audited at every point, never moved. And one account taxed them all: warmth — friction’s quiet commission, payable on every ride.”