Nature of science: a debate

Class 12 Physics, unit 26, book pages 211–220

This unit looks at the nature of science itself rather than at new formulas. It asks why many physicists see beauty in the mathematics of nature (elegance of simplicity and symmetry, in equations such as $E = m c^{2}$ and $e^{iπ} + 1 = 0$ ) and why that belief can mislead, how society debates whether to search for aliens and whether to continue research on subatomic particles after nuclear weapons, and how Newton's cannonball thought experiment explained orbits long before any real satellite existed.

Learning outcomes (4)

Explain, with examples, what thinkers who hold the view that there is inherent mathematical beauty in the natural world mean by (i) elegance of simplicity and (ii) symmetry. [P-12-G-01]
Explain, with an example, a counterargument to the claim that physical truths must be inherently mathematically elegant or display symmetry. [P-12-G-02]
Describe the main pros and cons in the debate about (i) whether humans should research whether there are aliens somewhere in the universe, and (ii) whether research should continue on uncovering the secrets of subatomic particles, given the advent of nuclear weapons. [P-12-G-03]
Explain how thought experiments helped convey important physics concepts that would have been impractical to investigate empirically: (i) Newton's cannonball. [P-11-G-04]

Mathematics: the language of science

The logic of shape, quantity and order is the subject of mathematics. It is used every day and everywhere: in computers, software and hardware, art, money, engineering, sports, and in ancient and modern architecture. Whether we enjoy it or not, mathematics is a necessary part of daily life.

Physics is written in the language of mathematics. A law such as $F = ma$ or $E = m c^{2}$ packs a great deal of experience into a few symbols. This unit looks at the nature of science itself: why many scientists find the mathematics of nature beautiful, why that belief can also mislead, how society debates what science should study, and how a thought experiment can teach physics that no real experiment could test at the time.

26.1Elegance of mathematics

Mathematics is an extremely elegant field, but seeing its elegance needs some knowledge. Its beauty lies in patterns, in the way numbers and shapes fit together, and in the traditional principles of symmetry and balance. The elegance of mathematics lies in how well everything fits together. Many people are frightened of numbers and never see this beauty; the book mentions a work called Mathematical Elegance whose author shows it through hundreds of approachable examples.

Mathematical elegance: The quality of a result, equation or proof that is simple and short, yet explains a great deal and fits neatly with other ideas.

Why science depends on mathematics

The natural sciences, engineering, health, economics, computer science and the social sciences all depend on mathematics.
The fundamentals of mathematics are not affected by scientific experiments, although mathematics is often used to model natural phenomena. A proved theorem stays true for ever; a physical model must still be tested by experiment.
Mathematics encourages logical reasoning and mental consistency, and it is an efficient way of developing mental discipline.
Understanding mathematics helps in other subjects, including natural science, social studies, music and the arts.
It is a highly brief language for expressing relationships of any kind quantitatively.

Mathematics also links with the arts. Poets and writers have explored the link between the richness of natural language and the exactness of mathematics, and artists use mathematics to create bold analogies and to set limits within which they express their creativity. The writer Jorge Luis Borges, inspired by mathematics, described the 'Aleph' as a point in space that contains all other points, from which every place on Earth can be seen at once from all angles.

26.1The magic of mathematics

Most people link mathematics with long equations, chalkboards covered with numbers and old tests. But mathematics is a universal language that bridges the vast scales of the universe: it describes the pattern of a seashell and the spirals of distant galaxies with equal precision. It is the thread that connects the Fibonacci patterns in sunflowers and pinecones with the ideas behind the technology we use every day. You experience it in the design of a building, in a beautiful melody and in the symmetry of a snowflake.

The Fibonacci sequence

Each number in the sequence is the sum of the two numbers before it:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \dots F_{n + 1} = F_{n} + F_{n - 1}

The scales of a pinecone and the seeds of a sunflower lie on two sets of spirals, and the numbers of spirals are usually neighbouring Fibonacci numbers, such as 8 and 13 in a pinecone or 34 and 55 in a sunflower. The ratio of neighbouring terms approaches the golden ratio, about $1.618$ .

Euler's identity: five numbers in one equation

Mathematicians are compared to poets: as a poet uses petals, colours and perfume to express feelings, an equation uses numbers and symbols as its 'petals'. The book's example is Euler's equation:

e^{iπ} + 1 = 0

$π \approx 3.14159$ is the ratio of the circumference of a circle to its diameter. It is fundamental to geometry and appears everywhere.
$e \approx 2.718281$ is used to describe anything that grows or decays in proportion to itself (for example radioactive decay, $N = N_{0} e^{- λ t}$ ). It is a fundamental idea in calculus and defines exponentiation; $e$ holds the same place in calculus that $π$ holds in geometry.
$i = - 1$ is the imaginary unit. It is not a real number and does not lie on the ordinary number line. Allowing square roots of $- 1$ produces the complex numbers, which have a real part and an imaginary part.
$0$ and $1$ are the most basic numbers of arithmetic.

The numbers $π$ , $e$ and $i$ come from quite different areas (geometry, calculus and algebra) and appear to have no connection. Euler's identity shows that geometry, calculus and complex numbers are intimately related. Complex numbers are the link between circular motion and arithmetic: they carry information about angles, rotations, cycles and periods, while real numbers tell us about quantities.

Derivation Euler's identity from Euler's formula

Aim. Show that $e^{iπ} + 1 = 0$ .

Assumptions.

Angles are in radians, and Euler's formula $e^{i θ} = cos θ + i sin θ$ is taken from mathematics.

Euler's formula (from mathematics) says that $e^{i θ}$ is the point at angle $θ$ on a circle of radius 1: its real part is $cos θ$ and its imaginary part is $sin θ$ .
$e^{i θ} = cos θ + i sin θ$
Put $θ = π$ , which is half a turn.
$e^{iπ} = cos π + i sin π$
Use $cos π = - 1$ and $sin π = 0$ .
$e^{iπ} = - 1 + 0 \cdot i = - 1$
Add 1 to both sides.
$e^{iπ} + 1 = 0$

Result

e^{iπ} + 1 = 0

Turning half-way round the unit circle takes the number 1 to $- 1$ . The identity joins the constant of growth ( $e$ ), the constant of circles ( $π$ ), the imaginary unit ( $i$ ) and the numbers 0 and 1 in one short statement, which is why it is often called the most beautiful equation in mathematics.

Euler's identity on the unit circle

−1.5

−1

−0.5

0.5

1.5

−1

−0.5

0.5

Re

Im

e^{i θ}

e^{i 0} = 1

e^{iπ} = - 1

θ

e^{i θ} = cos θ + i sin θ

e^{i θ}

is the point at angle

θ

on a circle of radius 1. Half a turn (

θ = π

) takes

1

- 1

, so

e^{iπ} = - 1

and

e^{iπ} + 1 = 0

26.1SymmetryOften examined

Symmetry (in mathematics): The balance and proportion that an object or geometric figure displays when one half reflects the other: specific points, lines or forms match one another exactly along a given axis.
Symmetry (general, in physics): Any change (transformation), such as a reflection, a shift or a rotation, after which an object or a law looks exactly the same.

People usually link symmetry with reflecting a picture in a mirror, but that is only one kind. Rotational symmetry, for example, means that an object's properties remain unchanged as it is rotated about an axis. The study of symmetry is essential to group theory, an important area of algebra with applications in computer graphics, chemistry and crystallography.

Four kinds of symmetry

Kind	What stays the same	Examples
Reflection (mirror) symmetry	The shape looks the same after reflection in a line: one half is the mirror image of the other.	A butterfly, a square about its diagonals
Translational symmetry	A pattern looks the same after it is slid (translated) through a fixed distance.	A row of identical triangles, wallpaper, the atoms of a crystal
Rotational symmetry	The shape looks the same after rotation through some angle about a centre.	A regular hexagon (every $6 0^{\circ}$ ), a starfish (every $7 2^{\circ}$ ), a snowflake
Point symmetry	Every point $P$ has a matching point $P^{'}$ at the same distance on the opposite side of a central point O (a half-turn about O).	The letters S, Z and N; the pairs of F shapes and stars placed about a central point in the book's figure

The four kinds of symmetry named in the book: (i) reflection about a mirror line, (ii) translation by equal steps, (iii) rotation (a hexagon looks the same after every

6 0^{\circ}

turn), (iv) point symmetry about the centre O.

Applications of symmetry

Architecture: structurally sound and pleasing buildings result from architects using symmetry.
Art and design: artists use symmetry to give harmony and balance to their work.
Nature: the radial symmetry of a starfish and the hexagonal symmetry of honeycombs are two of many examples.
Music: composers use symmetry in melodies and harmonies to arouse feelings and give enjoyable sound.
Science: the symmetry of crystals and molecules, and the symmetry of physical laws themselves.

Symmetry makes physics problems simpler

According to the mathematician Ivan Loseu of Northeastern University, the three-dimensional world makes many problems hard, and the best route to a solution is to reduce the number of variables. The motion of the Earth around the Sun shows how:

In three dimensions, the Earth's position needs three variables.
The Earth never leaves one plane (the plane of its orbit), so Newtonian physics reduces this to two variables.
The force of gravity depends only on the distance between the Sun and the Earth, not on the direction. This symmetry reduces the key variable to one, the distance $r$ . That is why physicists can track the Earth's elliptical journey using the properties of the gravitational force.

F = \frac{GM m}{r ^{2}}

$F$ depends only on the distance $r$ . Rotating the whole picture about the Sun leaves the law unchanged (rotational symmetry).

26.1What scientists say about mathematics

Scientist	What they said
Albert Einstein, German theoretical physicist	'Pure mathematics is, in its way, the poetry of logical ideas.'
Katherine Johnson, African-American mathematician	'We will always have STEM with us. Some things will drop out of the public eye and go away, but there will always be science, engineering, and technology. And there will always, always be mathematics.'
Richard Courant, German-American mathematician	'Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality.'
Sofia Kovalevskaya, Russian mathematician	'It is impossible to be a mathematician without being a poet in soul.'

26.1Elegance of mathematical poetry: four famous equationsOften examined

Some equations are called beautiful because a few symbols hold a huge amount of truth. The book gives four examples.

Pythagoras' theorem

Fig. 26.1: Pythagoras' theorem for a right-angled triangle with shorter sides

a

b

and hypotenuse

c

a^{2} + b^{2} = c^{2}

$a$ , $b$ : the two shorter sides of a right-angled triangle, $c$ : the hypotenuse.

This equation gives the relationship between the sides of every right-angled triangle, a fundamental idea in geometry. Its simplicity and universal validity are reflected in rocket flight (combining perpendicular velocities), building design (a $3$ , $4$ , $5$ triangle gives an exact right angle) and many other uses.

Mass-energy equivalence

E = m c^{2}

$E$ : energy, $m$ : mass, $c = 3 \times 1 0^{8} m s^{- 1}$ : speed of light, a constant, which is squared.

Arguably the best-known equation in physics. It connects mass, energy and the speed of light, and reveals that mass and energy are inter-convertible, with beauty and conciseness. Because $c^{2}$ is huge, a tiny mass is equivalent to an enormous energy: $1 g$ of mass corresponds to $E = (1 0^{- 3}) (3 \times 1 0^{8})^{2} = 9 \times 1 0^{13} J$ . This is the energy source of nuclear reactors and of the Sun.

Planck-Einstein relation

E = h f = \frac{h c}{λ}

$h = 6.63 \times 1 0^{- 34} J s$ : Planck's constant, $f$ : frequency (the book writes $ν$ ), $λ$ : wavelength.

A fundamental formula of quantum physics: a quantum of energy $E$ (a photon) equals Planck's constant times the frequency of the oscillator. Planck's idea of quantised energy is used to explain how light liberates electrons from a photovoltaic material and produces an electric current. For light of wavelength $500 nm$ , $E = \frac{( 6.63 \times 1 0 ^{- 34} ) ( 3 \times 1 0 ^{8} )}{5 \times 1 0 ^{- 7}} = 3.98 \times 1 0^{- 19} J \approx 2.5 eV$ .

Euler's identity

e^{iπ} + 1 = 0

This equation establishes an amazingly simple relationship between five of the most important numbers in mathematics, $0$ , $1$ , $π$ , $e$ and $i$ . It unites separate branches of mathematics in harmony, like a moving poem.

Equation	Why it is called elegant
$a^{2} + b^{2} = c^{2}$	Three letters describe every right-angled triangle.
$E = m c^{2}$	Links two quantities once thought separate: mass and energy.
$E = h f$	One constant links the energy of light to its frequency.
$e^{iπ} + 1 = 0$	Joins five basic numbers from different branches of mathematics.

26.1Beauty in physics: elegance of simplicity and symmetryOften examined

Many great physicists hold the view that there is an inherent mathematical beauty in the natural world. Paul Dirac believed that a correct physical law should have mathematical beauty, and Einstein searched for the simplest possible equations. When such thinkers call a law beautiful, they mean mainly two things: elegance of simplicity and symmetry.

(i) Elegance of simplicity

A law is elegant when a few simple principles, or one short equation with nothing unnecessary, explain a very wide range of facts, and especially when it unites things that looked unrelated. Choosing the simplest explanation that fits all the facts is also called Occam's razor.

Newton's law of gravitation, $F = \frac{G m _{1} m _{2}}{r ^{2}}$ : one short law explains a falling apple, the orbit of the Moon, the tides and the motion of the planets. The heavens and the Earth obey the same rule.
Maxwell's equations: four equations explain all of electricity and magnetism and show that light is an electromagnetic wave.
** $E = m c^{2}$ :** three symbols link mass and energy.
** $E = h f$ :** one constant links the energy of light to its frequency.

(ii) Symmetry

A law has symmetry when it stays the same after a change: moving the experiment to another place or time, turning it round, or reflecting it. Thinkers see symmetry as a sign of deep truth because it makes laws simple and powerful, and it has even predicted discoveries.

Gravity depends only on distance: the Earth's orbit can be described with the single variable $r$ (Loseu's example).
Electricity and magnetism: a changing magnetic field produces an electric field, and a changing electric field produces a magnetic field. From this matching (symmetric) pair of effects Maxwell predicted electromagnetic waves travelling at the speed of light.
Nature: the six-fold symmetry of snowflakes and honeycombs and the regular patterns of crystals come from simple laws acting the same way in every direction.

26.1A counterargument: physical truths need not be elegantOften examined

The claim: physical truths must be mathematically elegant or symmetric, so a beautiful theory is likely to be true. The counterargument: nature is under no obligation to be beautiful. A theory is accepted because its predictions agree with careful measurements, not because it is elegant. Beautiful ideas have often turned out wrong, and many true laws are untidy or break symmetry.

Example 1: Kepler's ellipses

Since ancient Greek times astronomers believed that heavenly bodies must move in perfect circles, the most symmetric curve. In 1596 Johannes Kepler proposed a beautiful model in which the orbits of the six known planets fitted around the five perfectly symmetric solids of geometry (the Platonic solids). But when he analysed Tycho Brahe's very accurate observations of Mars, no circle fitted the data: the errors were small but real. In 1609 Kepler accepted that the orbit is an ellipse with the Sun at one focus, a less symmetric shape. The measurements beat the beauty. (The book itself notes that the Earth follows an elliptical path.)

Circle versus ellipse (eccentricity exaggerated)

−1.5

−1

−0.5

0.5

−1

−0.5

0.5

x

y

Sun

circle: the symmetric guess

ellipse: what the data showed

Both orbits have the same size. Kepler expected a perfectly symmetric circle centred on the Sun, but Tycho Brahe's accurate data for Mars fitted only an ellipse with the Sun at one focus (here eccentricity

0.5

). The real orbit of Mars is much closer to a circle (eccentricity about

0.09

) than this drawing.

Conclusion. Elegance and symmetry have guided physicists to many great discoveries, but they are a guide, not a proof. Nature is sometimes elegant and sometimes untidy, and only measurement can tell which.

26.2Are aliens real?

The prospect of life beyond Earth has long fascinated human curiosity. Since people discovered that Mars and the other planets are different from the points of light of the stars, they have wondered about otherworldly life. The question of whether we are alone in the universe is still unanswered, even though modern technology lets us study other worlds up close and even detect planets orbiting other stars.

Many people believe in aliens, but this belief is mostly based on movies and reports of UFOs (unidentified flying objects), which are not scientific evidence. Science approaches the question with evidence: it asks where life could exist and looks for signs of it.

Extraterrestrial life (aliens): Life that exists beyond the Earth.
Exoplanet: A planet that orbits a star other than the Sun.
UFO: Unidentified flying object: something seen in the sky that the observer cannot identify. It is not evidence of aliens.

Where might aliens exist?

Mars: NASA's exploration programme has shown that the red planet was once warm and wet and may have supported life. No evidence of life has been found yet, but tiny (microbial) life might still be present.
Icy moons: several moons of Jupiter and Saturn have liquid water beneath their ice crusts. Europa (a moon of Jupiter) and Enceladus (a moon of Saturn) appear to have underground oceans that could support life.
Exoplanets: thousands of planets have been discovered around other stars, and some lie at the right distance from their star for liquid water. The number of possible homes for life keeps increasing, and this is a major field of future research.

26.2Debate: should humans research the existence of aliens?Often examined

The book lists the main arguments on each side of this debate.

Pros (for the research)	Cons (against the research)
Expanding human knowledge: it can give a deeper understanding of the universe, its origins and the possibility of life beyond Earth.	Resource allocation: it needs large funding and resources that could be used for pressing problems on Earth, such as poverty, disease and climate change.
Potential benefits: discovering alien life could lead to new technologies, resources and insights that benefit humanity.	Low probability of success: the chance of finding alien life is unknown, and some argue it is too low to justify the investment.
Inspiring future generations: the search inspires young people to choose careers in science, technology, engineering and mathematics (STEM).	Risk of unintended consequences: meeting alien life could bring contamination (of the Earth or of other worlds) or even conflict.

A balanced view. Much of this research is shared with other sciences (space technology, telescopes, the biology of life in extreme places), so it gives useful results even if no alien life is found, and strict planetary-protection rules reduce the risk of contamination. Against this, every rupee spent must be weighed against urgent needs on Earth.

26.2Debate: should research on subatomic particles continue?Often examined

Research on the atom and its particles led to nuclear energy and to new medicine, but also to nuclear weapons. So people ask whether this research should continue.

Pros (continue the research)	Cons (limit the research)
Advancing fundamental knowledge: it gives a deeper understanding of the fundamental laws of physics and the nature of matter.	Nuclear weapons: the discovery of subatomic particles and the development of nuclear physics led to nuclear weapons, which pose a serious threat to global security.
Potential applications: discoveries lead to breakthroughs in medicine (PET scans, radiotherapy), energy (nuclear power) and materials science.	Ethical concerns: continuing research might lead to new, potentially devastating technologies.
Inspiring innovation: the pursuit of this knowledge drives innovation and new technologies.	Opportunity cost: the huge resources needed (large particle accelerators) could be spent on other areas of science or on pressing global problems.

History shows both sides. The discovery of nuclear fission in 1938 led within seven years to the atomic bombs of 1945. The same physics gives nuclear power stations, cancer treatment and medical imaging (Unit 25), and the World Wide Web was invented at CERN, a particle physics laboratory, in 1989 so that physicists could share data.

A common conclusion. Knowledge itself is neither good nor bad; people and governments decide how it is used. Stopping research would lose its benefits without removing the weapons that already exist. The better path is to continue research while controlling misuse through ethics, international treaties (such as the Nuclear Non-Proliferation Treaty of 1968) and public debate. The debate shows the complex trade-offs and ethical considerations involved in pursuing scientific knowledge.

26.3Thought experiments

Thought experiment: An experiment carried out in the imagination, using known laws and logical reasoning, to explore an idea or explain a concept when a real experiment is impossible, too costly or impractical.

The German name is Gedankenexperiment. A thought experiment needs no apparatus: it follows the laws of physics step by step to see what they predict in an ideal situation. It is used to explain a concept clearly, to test whether a theory is consistent, or to reach situations that no laboratory can create.

26.3Thought experiment: Newton's cannonballOften examined

It was common knowledge that objects fall to the ground, but no one was sure why. The term 'gravitas', which simply means 'having weight', was applied to this unexplained effect. Newton did not discover gravity; he explained it with a clear mathematical law that could be tested and that tied many effects together. The apple and the Moon are both pulled towards the Earth by gravity. The Moon also moves sideways, which accounts for its different motion and is the reason it does not land on the Earth.

Newton's two questions

Why doesn't the Moon fall from the sky and land on the Earth, given that all objects experience a gravitational acceleration towards the Earth?
Why does the Moon move in a circle instead of a straight line if, as Galileo said, objects move with constant speed and direction until acted upon by an external force?

Newton understood that because of the combination of its orbital motion and its inertia, the Moon neither falls onto the Earth nor flies away, even though it has an acceleration towards the Earth.

The thought experiment

Newton imagined a cannon on a towering mountain (high enough to be above the air, so there is no air resistance) that fires a ball horizontally, Fig. 26.5. In time $t$ the ball moves sideways and, at the same time, falls under gravity:

x = v t and y = \frac{1}{2} g t^{2}

$x$ : horizontal distance, $v$ : horizontal speed of the ball, $y$ : vertical distance fallen, $g$ : acceleration due to gravity, $t$ : time.

Because the Earth is round, its surface curves away below the ball. If Newton chose just the right speed, the ball's path would curve at exactly the same rate as the Earth's surface, and the ball would always stay at the same height above the Earth. It keeps falling towards the Earth but never gets closer: it is in orbit.

Newton's cannonball (after Figs. 26.5 to 26.8, drawn to scale for a very high mountain). The faster the ball, the farther it goes; at the orbital speed it circles the Earth.

Results

Speed of the ball	What happens	Book figure
Less than the orbital speed ('too slow')	It falls back to the Earth; a faster ball lands farther away.	Fig. 26.6
Equal to the orbital speed ('just right')	It stays in a circular orbit at the same height.	Fig. 26.7
Greater than the orbital speed ('too fast')	It moves away from the Earth: it follows an ellipse, and at or above the escape speed it leaves the Earth for ever.	Fig. 26.8

In orbit there is a balance between the ball's inertia, which makes it keep going in a straight line away from the Earth, and the acceleration due to gravity, which pulls it towards the Earth's centre. The ball accelerates even though its speed is constant, because the direction of its velocity keeps changing. Newton related this acceleration to the speed $v$ and the radius $r$ of the orbit (measured from the Earth's centre):

a = \frac{v ^{2}}{r}

Centripetal acceleration. Newton found it for the cannonball, but it holds for any object moving in a circle; the acceleration for the cannonball comes from the Earth's gravity.

Derivation Orbital speed from Newton's cannonball

Aim. Find the speed $v$ at which a ball fired horizontally near the Earth's surface stays in orbit.

Assumptions.

There is no air resistance, and the height of the mountain is small compared with the Earth's radius $R$ .
$g$ is constant over the short distance considered.

The ball leaves P horizontally with speed

v

. In time

t

it would reach Q along the tangent, but it falls

\frac{1}{2} g t^{2}

. The Earth's surface drops by

h = SQ

below the tangent, and

OQ = R + h

In a short time $t$ the ball moves a horizontal distance $x = v t$ along the tangent PQ, and at the same time falls a vertical distance $y = \frac{1}{2} g t^{2}$ .
$x = v t, y = \frac{1}{2} g t^{2}$
Meanwhile the Earth's surface drops below the tangent by a height $h = SQ$ . Triangle OPQ has a right angle at P (a tangent is perpendicular to the radius), so Pythagoras' theorem gives:
$(R + h)^{2} = R^{2} + x^{2}$
Expand. The drop $h$ is tiny compared with $R$ , so $h^{2}$ is negligible.
$R^{2} + 2 R h + h^{2} = R^{2} + x^{2} \Rightarrow h \approx \frac{x ^{2}}{2 R}$
For the ball to stay at the same height above the ground, it must fall exactly as far as the ground drops away: $y = h$ .
$\frac{1}{2} g t^{2} = \frac{( v t ) ^{2}}{2 R}$
Cancel $t^{2} /2$ and solve for $v$ .
$g = \frac{v ^{2}}{R} \Rightarrow v = g R$
Put in $g = 9.8 m s^{- 2}$ and $R = 6.4 \times 1 0^{6} m$ .
$v = (9.8) (6.4 \times 1 0^{6}) = 7.9 \times 1 0^{3} m s^{- 1}$

Result

v = g R \approx 7.9 km s^{- 1}

At about $8 km s^{- 1}$ the ball falls $4.9 m$ in the first second, and over the $7.9 km$ it travels the Earth's surface also curves down by $4.9 m$ , so the ball never gets closer to the ground. The step $g = v^{2} / R$ says that in orbit the whole acceleration due to gravity acts as the centripetal acceleration $a = v^{2} / r$ ; it agrees with $v_{o} = GM / R$ from Unit 15, since $g = GM / R^{2}$ .

Period of the cannonball's orbit

How long would Newton's cannonball take to go once round the Earth at the orbital speed

v = 7.9 \times 1 0^{3} m s^{- 1}

? Take

R = 6.4 \times 1 0^{6} m

In one orbit the ball covers the circumference $2 π R$ at constant speed $v$ .
$T = \frac{2 π R}{v}$
Substitute.
$T = \frac{2 π ( 6.4 \times 1 0 ^{6} )}{7.9 \times 1 0 ^{3}} = 5.1 \times 1 0^{3} s$
Convert to minutes.
$T = \frac{5.1 \times 1 0 ^{3}}{60} \approx 85 min$

T \approx 5.1 \times 1 0^{3} s \approx 85 min

Why a thought experiment, and what it taught

Impractical to test: no cannon could fire a ball at about $8 km s^{- 1}$ (about $28000 km h^{- 1}$ ), no mountain rises above the air, and air resistance would quickly slow the ball. In Newton's time the idea could only be tested in the mind.
Gravity is universal: the same gravity that makes an apple fall keeps the Moon in orbit. The Moon is simply a 'cannonball' moving sideways fast enough.
An orbit is continuous free fall: a satellite is always falling towards the Earth but moves sideways fast enough to keep missing it.
Projectile motion and circular motion are linked: $x = v t$ and $y = \frac{1}{2} g t^{2}$ lead to the centripetal acceleration $a = v^{2} / r$ .
It predicted artificial satellites: about 270 years later, in 1957, Sputnik 1 became the first real 'Newton's cannonball'.

Formula sheet

Euler's identity

e^{i\pi}+1=0

e\approx2.718, \pi\approx3.142, i=\sqrt{-1}

no units (pure numbers)

Fibonacci sequence

F_{n+1}=F_{n}+F_{n-1}

F_n: the nth term; 1, 1, 2, 3, 5, 8, \ldots

no units

Pythagoras' theorem

a^{2}+b^{2}=c^{2}

a, b: shorter sides, c: hypotenuse of a right-angled triangle

any unit of length

Mass-energy equivalence

E=mc^{2}

m: mass, c=3\times10^{8}\,\mathrm{m\,s^{-1}}

Units: E in \mathrm{J}, m in \mathrm{kg}

Planck-Einstein relation

E=hf=\dfrac{hc}{\lambda}

h=6.63\times10^{-34}\,\mathrm{J\,s}, f: frequency, \lambda: wavelength

Unit: \mathrm{J}

Law of gravitation

F=\dfrac{GMm}{r^{2}}

G=6.67\times10^{-11}\,\mathrm{N\,m^{2}\,kg^{-2}}, M, m: masses, r: distance between centres

Unit: \mathrm{N}

Horizontal distance of the cannonball

x=vt

v: horizontal speed, t: time

Unit: \mathrm{m}

Vertical distance fallen

y=\dfrac{1}{2}gt^{2}

g=9.8\,\mathrm{m\,s^{-2}}, t: time

Unit: \mathrm{m}

Centripetal acceleration

a=\dfrac{v^{2}}{r}

v: speed, r: radius of the circular path

Unit: \mathrm{m\,s^{-2}}

Orbital speed near the Earth's surface

v=\sqrt{gR}\approx7.9\,\mathrm{km\,s^{-1}}

g=9.8\,\mathrm{m\,s^{-2}}, R=6.4\times10^{6}\,\mathrm{m}

Unit: \mathrm{m\,s^{-1}}

Period of a low orbit

T=\dfrac{2\pi R}{v}\approx85\,\mathrm{min}

R: radius of the orbit, v: orbital speed

Unit: \mathrm{s}

Escape speed (correction to the book)

v_{e}=\sqrt{2gR}\approx11.2\,\mathrm{km\,s^{-1}}

g: surface value, R: Earth's radius

Unit: \mathrm{m\,s^{-1}}

Key terms

Mathematics: The logic of shape, quantity and order; the language in which the laws of physics are written.
Mathematical elegance: The quality of an equation or proof that is simple and short, yet explains a great deal and fits neatly with other ideas.
Elegance of simplicity: The idea that a true law explains a wide range of facts with a few simple principles and nothing unnecessary.
Occam's razor: The rule of choosing the simplest explanation that fits all the facts.
Symmetry: The balance and proportion of a figure when one half reflects the other; more generally, any transformation after which an object or a law looks the same.
Reflection symmetry: A figure looks the same after reflection in a mirror line.
Translational symmetry: A pattern looks the same after it is slid through a fixed distance.
Rotational symmetry: An object's properties remain unchanged when it is rotated about an axis through some angle.
Point symmetry: Every point has a matching point at the same distance on the opposite side of a central point.
Group theory: The area of algebra that studies symmetry; used in computer graphics, chemistry and crystallography.
Fibonacci sequence: The sequence 1, 1, 2, 3, 5, 8, \ldots in which each term is the sum of the two before it; seen in sunflowers and pinecones.
Euler's identity: e^{i\pi}+1=0, which links the five numbers 0, 1, \pi, e and i.
Imaginary unit: i=\sqrt{-1}; it does not lie on the real number line and is the basis of complex numbers.
Counterargument: An argument against a claim; here, that physical truths need not be elegant or symmetric, because experiment, not beauty, decides.
Extraterrestrial life: Life beyond the Earth (aliens).
Exoplanet: A planet orbiting a star other than the Sun.
UFO: Unidentified flying object; not scientific evidence of aliens.
STEM: Science, technology, engineering and mathematics.
Opportunity cost: What is given up when resources are spent on one thing instead of another.
Thought experiment: An experiment carried out in the imagination with known laws and logic, used when a real experiment is impractical.
Newton's cannonball: Newton's thought experiment of a ball fired horizontally from a very high mountain, showing that a fast enough projectile goes into orbit.
Inertia: The tendency of a body to keep moving in a straight line at constant speed unless a force acts on it.
Orbital speed: The speed at which a body moves in a circular orbit; near the Earth's surface v=\sqrt{gR}\approx7.9\,\mathrm{km\,s^{-1}}.
Centripetal acceleration: The acceleration a=v^{2}/r towards the centre of a circular path.
Escape speed: The least speed needed to leave the Earth for ever, \sqrt{2gR}\approx11.2\,\mathrm{km\,s^{-1}} from the surface.

Practice questions

The book has no end-of-unit exercise for this unit. These questions are written for revision in FBISE style (MCQs, short questions and comprehensive questions) and are not from the book.

Euler's identity e^{i\pi}+1=0 connects five important numbers. They are:

A0,\ 1,\ \pi,\ e,\ iAnswer
B0,\ 1,\ \pi,\ c,\ h
C1,\ 2,\ \pi,\ e,\ g
D0,\ 1,\ c,\ h,\ G

Why

The identity contains exactly e, i, \pi, 1 and 0. The other options mix in physical constants (c, h, g, G), which do not appear in it.

A row of identical triangles that looks exactly the same when it is slid along the row by one spacing shows:

Areflection symmetry
Btranslational symmetryAnswer
Crotational symmetry
Dpoint symmetry

Why

Sliding (translating) a pattern through a fixed distance and finding it unchanged is translational symmetry, as in the book's row of triangles. Reflection needs a mirror line and rotation needs a turn about a centre.

In Newton's cannonball thought experiment, a ball fired horizontally with a speed less than the orbital speed will:

Amove in a circular orbit
Bfall back to the EarthAnswer
Cescape from the Earth
Dmove in a straight line

Why

Too slow: the ball falls \tfrac12gt^{2} faster than the Earth's surface curves away, so it lands on the Earth (Fig. 26.6). Only at the orbital speed does it stay in a circular orbit.

The orbital speed of Newton's cannonball close to the Earth's surface (g=9.8\,\mathrm{m\,s^{-2}}, R=6.4\times10^{6}\,\mathrm{m}) is about:

A3.1\,\mathrm{km\,s^{-1}}
B7.9\,\mathrm{km\,s^{-1}}Answer
C11.2\,\mathrm{km\,s^{-1}}
D340\,\mathrm{m\,s^{-1}}

Why

v=\sqrt{gR}=\sqrt{(9.8)(6.4\times10^{6})}=7.9\times10^{3}\,\mathrm{m\,s^{-1}}

Option C, 11.2\,\mathrm{km\,s^{-1}}, is the escape speed \sqrt{2gR}, not the orbital speed.

Which of the following is given in the book as a 'con' in the debate on continuing research on subatomic particles?

AAdvancing fundamental knowledge
BPotential applications in medicine and energy
CNuclear weaponsAnswer
DInspiring innovation

Why

Nuclear weapons, a threat to global security, are the first con listed. The other three options are the book's pros.

Which of these moons is thought to have liquid water beneath its icy crust?

AThe Earth's Moon
BPhobos (a moon of Mars)
CEnceladus (a moon of Saturn)Answer
DDeimos (a moon of Mars)

Why

The book names Europa (Jupiter) and Enceladus (Saturn) as frozen worlds with possible underground water. The Earth's Moon and the two small moons of Mars have no such oceans.

73 marks

What do thinkers mean by the 'elegance of simplicity' in physics? Give one example.

Solution

Elegance of simplicity means that one short law or equation, with nothing unnecessary, explains a very wide range of facts, often uniting things that seemed unrelated. Thinkers who believe in the mathematical beauty of nature take such simplicity as a sign that a law is true.

Example: Newton's law of gravitation F=\dfrac{Gm_1m_2}{r^{2}}. One short formula explains why an apple falls, why the Moon orbits the Earth, why tides occur and how the planets move round the Sun. Other examples are E=mc^{2}, which links mass and energy in three symbols, and Maxwell's four equations, which cover all of electricity, magnetism and light.

83 marks

What is symmetry? How does symmetry allow the Earth's orbit around the Sun to be described with only one variable?

Solution

Symmetry is the balance of a figure when one half reflects the other; more generally, it is any transformation (reflection, rotation, shift) after which an object or a law stays the same.

In three dimensions the Earth's position needs three variables. Its orbit stays in one plane, so two are enough. The gravitational force F=\dfrac{GMm}{r^{2}} depends only on the distance r from the Sun, not on the direction, so turning the whole picture about the Sun leaves the law unchanged (rotational symmetry). The problem therefore reduces to one variable, r. As Loseu put it, you can change the image, but the physical law doesn't change.

93 marks

Give a counterargument, with an example, to the claim that physical truths must be mathematically elegant or symmetric.

Solution

Counterargument: a theory is accepted because its predictions agree with measurements, not because it is beautiful. Nature need not be elegant, and beautiful ideas have often been wrong.

Example: for about 2000 years astronomers believed planets must move in perfect circles, the most symmetric curve, and Kepler first built a beautiful model of circular orbits around the Platonic solids. Tycho Brahe's accurate measurements of Mars did not fit any circle, and in 1609 Kepler accepted a less symmetric ellipse with the Sun at one focus. Another example: mirror symmetry was expected in all laws, but Wu's cobalt-60 experiment (1957) showed that the weak force breaks it.

103 marks

What is a thought experiment? Why did Newton investigate the cannonball as a thought experiment and not as a real experiment?

Solution

A thought experiment is an experiment carried out in the imagination, using known laws and logical reasoning, to explain a concept when a real experiment is impossible or impractical.

Newton's cannonball could not be tested for real in his time: the ball would need a speed of about 8\,\mathrm{km\,s^{-1}}, far beyond any cannon; no mountain rises above the air; and air resistance would quickly slow the ball. By reasoning with x=vt and y=\tfrac12gt^{2} on a curved Earth, Newton showed that a fast enough ball keeps falling around the Earth, so the Moon and an apple obey the same gravity. Real satellites confirmed this in 1957.

117 marks

(a) Explain, with examples, what thinkers who believe in the mathematical beauty of nature mean by elegance of simplicity and by symmetry. (b) Give a counterargument, with an example, to the claim that physical truths must be elegant or symmetric.

Solution

Mathematical beauty in nature

Many physicists, such as Dirac and Einstein, hold that the laws of nature have an inherent mathematical beauty. 'Pure mathematics is, in its way, the poetry of logical ideas' (Einstein). By beauty they mean mainly elegance of simplicity and symmetry.

(a)(i) Elegance of simplicity

A law is elegant when one short equation, with nothing unnecessary, explains a very wide range of facts and unites things that looked unrelated.

Newton's law of gravitation F=\dfrac{Gm_1m_2}{r^{2}} explains the falling apple, the Moon's orbit, the tides and the planets with one rule.
E=mc^{2} links mass and energy in three symbols; E=hf links the energy of light to its frequency.
Maxwell's four equations explain electricity, magnetism and light together.

(a)(ii) Symmetry

A law is symmetric when it stays the same after a change such as a shift in place or time, a rotation or a reflection.

Gravity depends only on distance, so the Earth's orbit can be described with the single variable r (Loseu's example): 'you can change the image, but the physical law doesn't change'.
Symmetry of the laws in time and in space gives conservation of energy and of momentum (Noether's theorem).
Dirac's beautiful equation for the electron predicted the positron, discovered in 1932.
Snowflakes, honeycombs and crystals show symmetric patterns produced by simple laws.

(b) Counterargument

Beauty cannot prove a law; only agreement with experiment can. Nature is not obliged to be elegant, and beautiful ideas have often failed.

Example: Kepler's orbits. Astronomers long believed that planets must move in perfect circles, and Kepler first proposed a beautiful model of circular orbits nested around the five Platonic solids. Tycho Brahe's accurate data for Mars fitted no circle, and in 1609 Kepler accepted that orbits are ellipses with the Sun at one focus: less symmetric, but true.

Other examples: the weak nuclear force breaks mirror symmetry (Wu, 1957); constants such as G=6.67\times10^{-11}\,\mathrm{N\,m^{2}\,kg^{-2}} have untidy values; and elegant theories such as string theory remain unconfirmed.

Conclusion

Elegance and symmetry are powerful guides for finding new laws, but experiment is the final judge of what is true.

127 marks

Describe the main pros and cons in the debate about: (a) whether humans should research whether there are aliens somewhere in the universe; (b) whether research should continue on uncovering the secrets of subatomic particles, given the advent of nuclear weapons.

Solution

(a) Research on the existence of aliens

Mars was once warm and wet, icy moons such as Europa and Enceladus have hidden oceans, and thousands of exoplanets are known, so the question of whether we are alone can now be studied scientifically. The debate is whether the research is worth its cost.

Pros	Cons
Expanding knowledge: a deeper understanding of the universe, its origins and the possibility of life beyond Earth.	Resource allocation: the money could be used for poverty, disease and climate change on Earth.
Potential benefits: new technologies, resources and insights for humanity.	Low probability of success: the chance of finding alien life is unknown and may be too low to justify the cost.
Inspiring future generations: it attracts young people to STEM careers.	Unintended consequences: risk of contamination or conflict if alien life is met.

(b) Research on subatomic particles

Nuclear physics gave the world both nuclear power and medical imaging, and the atomic bomb (fission discovered in 1938, bombs used in 1945).

Pros	Cons
Advancing fundamental knowledge of the laws of physics and the nature of matter.	Nuclear weapons: the development of nuclear physics led to weapons that threaten global security.
Potential applications in medicine (PET, radiotherapy), energy (nuclear power) and materials science.	Ethical concerns: research might lead to new, devastating technologies.
Inspiring innovation: for example, the World Wide Web was invented at CERN in 1989.	Opportunity cost: huge accelerators use resources needed elsewhere.

Conclusion

Both debates involve trade-offs between the value of knowledge and its cost and risks. A balanced view is to continue both kinds of research at a sensible cost, share the benefits widely, and control the risks through ethics, planetary-protection rules and international treaties such as the Nuclear Non-Proliferation Treaty. Knowledge itself is neutral; how people use it decides whether it helps or harms.

137 marks

Describe Newton's cannonball thought experiment. What happens when the ball is fired with a speed less than, equal to and greater than the orbital speed? Show that the orbital speed near the Earth's surface is v=\sqrt{gR}, and explain what the experiment taught.

Solution

The thought experiment

Newton asked why the Moon does not fall onto the Earth although gravity pulls it. He imagined a cannon on a very high mountain, above the air, firing a ball horizontally. In time t the ball moves sideways x=vt and falls y=\tfrac12gt^{2}. Because the Earth is round, its surface curves away below the ball.

Faster balls land farther away; at the orbital speed the ball circles the Earth.

Results

Speed less than orbital speed: the ball falls back to the Earth; the faster it is fired, the farther it lands.
Speed equal to orbital speed: the ball's path curves exactly like the Earth's surface, so it stays at the same height and moves in a circular orbit.
Speed greater than orbital speed: the ball moves away from the Earth on an ellipse, and at or above the escape speed (\sqrt{2gR}\approx11.2\,\mathrm{km\,s^{-1}}) it leaves the Earth for ever.

Orbital speed

Derivation Orbital speed from Newton's cannonball

Aim. Find the speed v at which a ball fired horizontally near the Earth's surface stays in orbit.

Assumptions.

There is no air resistance, and the height of the mountain is small compared with the Earth's radius R.
g is constant over the short distance considered.

The ball leaves P horizontally with speed v. In time t it would reach Q along the tangent, but it falls \tfrac12gt^2. The Earth's surface drops by h=\mathrm{SQ} below the tangent, and \mathrm{OQ}=R+h.

In a short time t the ball moves a horizontal distance x=vt along the tangent PQ, and at the same time falls a vertical distance y=\tfrac{1}{2}gt^{2}.
x=vt,\qquad y=\tfrac{1}{2}gt^{2}
Meanwhile the Earth's surface drops below the tangent by a height h=\mathrm{SQ}. Triangle OPQ has a right angle at P (a tangent is perpendicular to the radius), so Pythagoras' theorem gives:
(R+h)^{2}=R^{2}+x^{2}
Expand. The drop h is tiny compared with R, so h^{2} is negligible.
R^{2}+2Rh+h^{2}=R^{2}+x^{2}\quad\Rightarrow\quad h\approx\dfrac{x^{2}}{2R}
For the ball to stay at the same height above the ground, it must fall exactly as far as the ground drops away: y=h.
\dfrac{1}{2}gt^{2}=\dfrac{(vt)^{2}}{2R}
Cancel t^{2}/2 and solve for v.
g=\dfrac{v^{2}}{R}\quad\Rightarrow\quad v=\sqrt{gR}
Put in g=9.8\,\mathrm{m\,s^{-2}} and R=6.4\times10^{6}\,\mathrm{m}.
v=\sqrt{(9.8)(6.4\times10^{6})}=7.9\times10^{3}\,\mathrm{m\,s^{-1}}

Resultv=\sqrt{gR}\approx7.9\,\mathrm{km\,s^{-1}}

At about 8\,\mathrm{km\,s^{-1}} the ball falls 4.9\,\mathrm{m} in the first second, and over the 7.9\,\mathrm{km} it travels the Earth's surface also curves down by 4.9\,\mathrm{m}, so the ball never gets closer to the ground. The step g=v^{2}/R says that in orbit the whole acceleration due to gravity acts as the centripetal acceleration a=v^{2}/r; it agrees with v_o=\sqrt{GM/R} from Unit 15, since g=GM/R^{2}.

What it taught

The same gravity that makes an apple fall keeps the Moon in orbit: gravity is universal.
An orbit is continuous free fall; inertia (sideways motion) and gravity are balanced, with centripetal acceleration a=v^{2}/r supplied by gravity.
It was impractical to test for real (a speed of 8\,\mathrm{km\,s^{-1}}, a mountain above the air, no air resistance), so reasoning in the mind was the only way; Sputnik 1 (1957) later confirmed it.

Exam focus

Explain, with examples, elegance of simplicity and symmetry in physics. (long)
SLO P-12-G-01; the most likely long question from this unit. No 2026 question came from Unit 26, so prepare from the SLOs.
Go to answer
Give a counterargument, with an example, to the claim that physical truths must be elegant or symmetric. (short)
SLO P-12-G-02; the book gives no example, so learn Kepler's ellipses or parity violation.
Go to answer
Name and explain the four kinds of symmetry, with examples. (short)
A neat 3-mark question built from the book's symmetry figure.
Go to answer
Pros and cons of researching the existence of aliens. (long)
SLO P-12-G-03 (i); learn the book's three pros and three cons.
Go to answer
Pros and cons of continuing research on subatomic particles, given nuclear weapons. (long)
SLO P-12-G-03 (ii); learn the book's three pros and three cons.
Go to answer
Describe Newton's cannonball thought experiment and its three results. (long)
SLO P-11-G-04; the only physics-heavy topic of the unit, with figures 26.5 to 26.8.
Go to answer
Show that the orbital speed near the Earth is v=\sqrt{gR}\approx7.9\,\mathrm{km\,s^{-1}}. (numerical)
Links the thought experiment to a=v^2/r and to orbital velocity in Unit 15.
Go to answer
Why are a^2+b^2=c^2, E=mc^2, E=hf and e^{i\pi}+1=0 called elegant? (short)
The book's four examples of mathematical poetry.
Go to answer