+++ /dev/null
-- Relative isotopic mass - mass of an isotope (compared to Carbon-12)
-- Atomic number (protons) is constant between isotopes - defines element.
-- Mass number - number of nucleons (defines isotope)
-- Relative atomic mass - average mass: $A_r = {{RIM \times abundance} \over 100}$
-- Relative atomic mass ($M_r$) - sum of relative atomic masses
-- Mole $6.02 \times 10^{23}$ (= no of atoms in Carbon-12) $=N_A$ (Avogadro's constant)
-- Molar mass - physical mass of 1 mol of a substance ($6.02 \times 10^{23}$ atoms)
- - Unit: grams / mole
-
-$$
-n = {m \over M}
+++ /dev/null
-# Organic Chemistry
-
-Organic compounds > 90% of known chemicals
-Carbon makes living systems possible
-
-Carbon forms 4 covalent bonds (4 valence pairs)
-Strong bonds with other non-metals
-Bonds with itself with single, double or triple
-Can form chains or rings
-
-Large number of compounds due to:
-- 4 valence pairs
-- single / double / triple bonds
-- cyclic (ring) structures
-
-
-## Hydrocarbons
-
-Hydrocarbons - only carbon and hydrogen
-If all C-C bonds are single, the hydrocarbon is saturated
-
-Hydrocarbons
-- linear (aliphatic)
-- - saturated (all single C-C bonds)
-- - - alkane ($C_n H_{2n+2}$)
-- - unsaturated
-- - - alkene ($C_n H_{2n}$)
-- - - alkyne ($C_n H_{2n-2}$)
--
-- cyclic (ring)
-- - saturated
-- - - e.g. cyclohexane
-- - unsaturated
-- - - e.g. benzene
-
-Carbon compounds with 4 branches are tetrahedral
-
-Compounds with $C-H$ are insoluble in water
-These compounds are almost non-polar due to similar electronegativities
-
-Only dispersion forces (between electrons on the outside of two molecules) hold each molecule together
-Dispersion forces increase with size (chain length), therefore MP and BP of straight molecules increases with length
-
-Branched molecules have lower MP and BP due to lower density (molecules are further apart due to their shape, so dispersion forces are weaker)
-
-Homologous series - formula for molecules (e.g. alkanes, alkenes, alkynes) where each member differs by some increment
-
-Structural isomers - same molecular formula, different configuration (structural formula)
-
-## Alkanes
-$$C_n H_{2n+2}$$
-- Single bonds between carbon atoms
-- ending in *-ane*
-
-methane $CH_4$
-ethane $C_2H_6$
-propane $C_3H_8$
-butane $C_4H_10$
-pentane $C_5H_12$
-hexane $C_6H_14$
-$\dots$
-
-Alkanes burn in oxygen forming $CO_2 + H_2O$ (oxidation / combustion reaction)
-
-Alkanes react with chlorine or fluorine (substitution reaction) - Cl or F replace H. Result is a haloalkane.
-
-## Alkenes
-
-$$C_nH_{2n}$$
-
-- C-C double bond
-- ending in *-ene*
-
-methene $CH_2$
-ethene $C_2H_4$
-propene $C_3H_6$
-butene $C_4H_8$
-pentene $C_5H_10$
-hexene $C_6H_12$
-$\dots$
-
-Alkenes burn in oxygen forming $H_2O + CO_2$.
-
-Alkenes react with hydrogen or a halogen (addition reaction) - additive breaks the double bond (catalyst is required).
-
-## Alkynes
-
-$$C_nH_{2n-2}$$
-
-- C-C triple bond
-- ending in *-yne*
-
-Alkynes undergo combustion (oxidation) and addition reactions (same as alkenes).
-
-## Rules for naming HCs
-
-1. Find longest chain of C atoms (incl. branches)
-2. Start at end nearest a branch or double/triple bond
-3. Name carbon atoms from end chosen
-4. Branches end with *-yl*
-5. When there are $\ge$ 2 branches, number of C atom is indicated before each branch, and names are in alphabetic order.
-6. When there are $\ge$ 2 identical branches, *di-*, *tri-*, and *tetra-* are used.
-
-## Combustion
-
-$$C_xH_y+z0_2 \to\ xCO_2+{y \over 2}H_2O$$
-$$z=x+{y \over 4}$$
-
-## Functional groups
-
-Atom(s), or a bond, which give a properties to an organic molecule.
-
-### Alcohols / hydroxyls
-
-- ---OH functional group
-- ends in *-ol*
-- alcohols have multiple isomers
-- combustible with oxygen
-- BP and MP are affected by H-bonding
-- **H-bonds are represented by dotted lines**
-
-### Carboxylic acids / carboxys
-
-- ---COOH functional group
-- ends in *-oic acid*
-- first portion of name (e.g. **methan**oic acid) refers to **total** number of carbons
-- weak acids (proton from O is donated)
-- high BP
-
-#### Esters
-- reacts with alcohols to form esters (condensation / esterification)
-- catalyst is $H_2SO_4$
-- low boiling points
-- intermolecular bonding is dipole-dipole and dispersion rather than H-bonding
-- ends in *-oate* rather than *-oic*
+++ /dev/null
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- code{white-space: pre-wrap;}
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-$endif$
-<p class="navbar">
-<span class="author"><a href="http://lorimer.id.au">Andrew Lorimer</a></span>
-<span class="sitenav"><a href="http://lorimer.id.au">home</a> // <a href="blog.html">blog</a> // <a href="photos.html">photos</a> // <a href="about.html">about</a></span>
-</p>
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-$endif$
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+++ /dev/null
-# Linear and Quadratic Functions
-
-## General binomial expansion
-
-$$(a+b)(c+d) = a(c+d) + b(c+d)$$
-$$= ac+ad+bc+bd$$
-
-## Perfect square expansion
-
-$$(a+b)^2=a^2+2ab+b^2$$
-
-## Difference of perfect squares
-
-$$ a^2-b^2=(a+b)(a-b)$$
-
-## Factorising monic quadratics
-
-$$x^2+bx+c = (x+e)(x-f)$$
-
-where
-$e \times f = c$
-$e+f = b$
-
-## Factorising general quadratics
-
-$ax^2+bx+c=(x+e)(x+f)$
-
-where
-$e \times f = a \times c$
-$e+f = b$
-
-## Completing the square
-
-$ax^2+bx+c=a(x + (b \over 2a))^2+(c-{b^2 \over 2a})^2$
-
-## Axis of symmetry
-
-${vert}_x={-b \over 2a}$
-
-$x={-b \over 2a}$
-(corresponding line)
+++ /dev/null
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- </head>
- <body class='markdown-preview'><h1>Linear and Quadratic Functions</h1>
-<h2>General binomial expansion</h2>
-<span class="math"><script type="math/tex; mode=display">(a+b)(c+d) = a(c+d) + b(c+d)
-</script></span>
-<span class="math"><script type="math/tex; mode=display">= ac+ad+bc+bd
-</script></span>
-<h2>Perfect square expansion</h2>
-<span class="math"><script type="math/tex; mode=display">(a+b)^2=a^2+2ab+b^2
-</script></span>
-<h2>Difference of perfect squares</h2>
-<span class="math"><script type="math/tex; mode=display">a^2-b^2=(a+b)(a-b)
-</script></span>
-<h2>Factorising monic quadratics</h2>
-<span class="math"><script type="math/tex; mode=display">x^2+bx+c = (x+e)(x-f)
-</script></span>
-<p>where<br>
-<span class="math"><script type="math/tex">e \times f = c</script></span><br>
-<span class="math"><script type="math/tex">e+f = b</script></span></p>
-<h2>Factorising general quadratics</h2>
-<span class="math"><script type="math/tex; mode=display">ax^2+bx+c=(x+e)(x+f)
-</script></span>
-<p>where<br>
-<span class="math"><script type="math/tex">e \times f = a \times c</script></span><br>
-<span class="math"><script type="math/tex">e+f = b</script></span></p>
-<h2>Completing the square</h2>
-<span class="math"><script type="math/tex; mode=display">ax^2+bx+c=a(x+{b \over 2a})^2+(c-{b^2 \over 2a})^2
-</script></span>
-<h2>Axis of symmetry</h2>
-<span class="math"><script type="math/tex; mode=display">{vert}_x={-b \over 2a}
-</script></span>
-<span class="math"><script type="math/tex; mode=display">x={-b \over 2a}
-</script></span>
-<p>(corresponding line)</p></body>
-</html>
+++ /dev/null
-<h1>Linear and Quadratic Functions</h1>
-<h2>General binomial expansion</h2>
-<span class="math"><script type="math/tex; mode=display">(a+b)(c+d) = a(c+d) + b(c+d)
-</script></span>
-<span class="math"><script type="math/tex; mode=display">= ac+ad+bc+bd
-</script></span>
-<h2>Perfect square expansion</h2>
-<span class="math"><script type="math/tex; mode=display">(a+b)^2=a^2+2ab+b^2
-</script></span>
-<h2>Difference of perfect squares</h2>
-<span class="math"><script type="math/tex; mode=display">a^2-b^2=(a+b)(a-b)
-</script></span>
-<h2>Factorising monic quadratics</h2>
-<span class="math"><script type="math/tex; mode=display">x^2+bx+c = (x+e)(x-f)
-</script></span>
-<p>where<br>
-<span class="math"><script type="math/tex">e \times f = c</script></span><br>
-<span class="math"><script type="math/tex">e+f = b</script></span></p>
-<h2>Factorising general quadratics</h2>
-<p><span class="math"><script type="math/tex">ax^2+bx+c=(x+e)(x+f)</script></span></p>
-<p>where<br>
-<span class="math"><script type="math/tex">e \times f = a \times c</script></span><br>
-<span class="math"><script type="math/tex">e+f = b</script></span></p>
-<h2>Completing the square</h2>
-<p><span class="math"><script type="math/tex">ax^2+bx+c=a(x + (b \over 2a))^2+(c-{b^2 \over 2a})^2</script></span></p>
-<h2>Axis of symmetry</h2>
-<p><span class="math"><script type="math/tex">{vert}_x={-b \over 2a}</script></span></p>
-<p><span class="math"><script type="math/tex">x={-b \over 2a}</script></span><br>
-(corresponding line)</p>
+++ /dev/null
-# Time Travel and Quantum Mechanics
-
-## What is time?
-- time is what clocks measure
- - measure time objectively, therefore does not help our subjective definition of time
-
-- $t$ - just like another quantitative variable (period of time, not absolute)
-
-
-- fourth dimension
-
-- these describe properties of it, not its implications or how it works
- - implications are generally determined through intuition
-
-
-- sense of time is inherent to all humans - every human has memory and some idea that time is observable through changes in space
-
- - computers have memory, do they have a concept of time?
-
-Therefore, we have:
-
-**1. Subjective time**
-- presumably everyone experiences this notion of time
-- has the property of lapsing (time approaches $\infty$)
-
-**2. Physical time**
-- represented by $t$
-- objective
-- quantitative (period, not absolute)
-
-> All paradoxes arise from the fact that time lapses / flows - we are basing time on subjective time, not physical time
-
-## Concepts of time in physics
-
-### Thermodynamic time arrow
-- second law of thermodynamics:
-- total entropy cannot decrease over time (randomness of particles)
-- Hence, time has a specific direction (approaching infinite entropy)
-- This can be modelled as $t=e$. If $e$ cannot be subtracted from, $t$ can only go one way.
-- entropy is an irreversible process
-- therefore, time travel backwards is impossible
-
-### Block time
-- time is fourth dimension - all dimensions are related
-- universe exists as a whole, in four dimensions, just like a chair can exist in three dimensions (plus time)
-- you cannot isolate one value in one of the three dimensions and define it by itself - it would be ridiculous to say that a point ($x=2,y=1,z=3$) can exist if y and z are not defined
-- therefore, past and future must also exist as a relation of four values ($x,y,z,t$)
-
-
-2. Are the past and future real?
-By using the present as a proxy for the past or future, we can simulate them, effectively nesting times inside times (a simpler idea of a time machine - normal time machines lose the idea of the present).
-
-3. Do events that occurred in the past / future exist?
-
-> What is the physical prompt to change to a different time?
-
-## Grandfather paradox
-
-- Grandfather is not necessary - could be the suicide paradox
-- Mathematical status - 1 bit (dead or alive):
- - $x=$ existence at 2 yrs, $y=$ existence at 20 yrs
- - $0=$ dead, $1=$ alive
-- $x=1$ (alive - assumed because you must exist in the first instance)
-
-
-**(case 1)** if $x=1$, then $y=1$ (assuming you don't die between) $\therefore x = y$
-**(case 2)** if $y=1$, then $x=0$ (because you killed yourself)
-This is not a contradiction - it simply states that $x=0 \lor 1$ (one *or* zero)
-
-**$\therefore$ both situations can exist (superposition) - suggesting multiple universe theory**
-
-- Implies a link to quantum mechanics: dead or alive, or between (superposition):
-- $1 \over 2$ probability of killing grandfather/self (two parallel realities - Murphy's law)
-
-
-### Many worlds interpretation
-
-- different branches of superposition are different universes. Perception jumps between universes.
-- time can be defined only by the physical state of things at the time - this is the only way it can be derived. therefore, by changing the physical state, you are creating a new "time (or timeline)
+++ /dev/null
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- <body class='markdown-preview'><h1>Time Travel and Quantum Mechanics</h1>
-<h2>What is time?</h2>
-<ul>
-<li>
-<p>time is what clocks measure</p>
-<ul>
-<li>measure time objectively, therefore does not help our subjective definition of time</li>
-</ul>
-</li>
-<li>
-<p><span class="math"><script type="math/tex">t</script></span> - just like another quantitative variable (period of time, not absolute)</p>
-</li>
-</ul>
-<ul>
-<li>
-<p>fourth dimension</p>
-</li>
-<li>
-<p>these describe properties of it, not its implications or how it works</p>
-<ul>
-<li>implications are generally determined through intuition</li>
-</ul>
-</li>
-</ul>
-<ul>
-<li>
-<p>sense of time is inherent to all humans - every human has memory and some idea that time is observable through changes in space</p>
-<ul>
-<li>computers have memory, do they have a concept of time?</li>
-</ul>
-</li>
-</ul>
-<p>Therefore, we have:</p>
-<p><strong>1. Subjective time</strong></p>
-<ul>
-<li>presumably everyone experiences this notion of time</li>
-<li>has the property of lapsing (time approaches <span class="math"><script type="math/tex">\infty</script></span>)</li>
-</ul>
-<p><strong>2. Physical time</strong></p>
-<ul>
-<li>represented by <span class="math"><script type="math/tex">t</script></span></li>
-<li>objective</li>
-<li>quantitative (period, not absolute)</li>
-</ul>
-<blockquote>
-<p>All paradoxes arise from the fact that time lapses / flows - we are basing time on subjective time, not physical time</p>
-</blockquote>
-<h2>Concepts of time in physics</h2>
-<h3>Thermodynamic time arrow</h3>
-<ul>
-<li>second law of thermodynamics:</li>
-<li>total entropy cannot decrease over time (randomness of particles)</li>
-<li>Hence, time has a specific direction (approaching infinite entropy)</li>
-<li>This can be modelled as <span class="math"><script type="math/tex">t=e</script></span>. If <span class="math"><script type="math/tex">e</script></span> cannot be subtracted from, <span class="math"><script type="math/tex">t</script></span> can only go one way.</li>
-<li>entropy is an irreversible process</li>
-<li>therefore, time travel backwards is impossible</li>
-</ul>
-<h3>Block time</h3>
-<ul>
-<li>time is fourth dimension - all dimensions are related</li>
-<li>universe exists as a whole, in four dimensions, just like a chair can exist in three dimensions (plus time)</li>
-<li>you cannot isolate one value in one of the three dimensions and define it by itself - it would be ridiculous to say that a point (<span class="math"><script type="math/tex">x=2,y=1,z=3</script></span>) can exist if y and z are not defined</li>
-<li>therefore, past and future must also exist as a relation of four values (<span class="math"><script type="math/tex">x,y,z,t</script></span>)</li>
-</ul>
-<ol start="2">
-<li>
-<p>Are the past and future real?<br>
-By using the present as a proxy for the past or future, we can simulate them, effectively nesting times inside times (a simpler idea of a time machine - normal time machines lose the idea of the present).</p>
-</li>
-<li>
-<p>Do events that occurred in the past / future exist?</p>
-</li>
-</ol>
-<blockquote>
-<p>What is the physical prompt to change to a different time?</p>
-</blockquote>
-<h2>Grandfather paradox</h2>
-<ul>
-<li>Grandfather is not necessary - could be the suicide paradox</li>
-<li>Mathematical status - 1 bit (dead or alive):
-<ul>
-<li><span class="math"><script type="math/tex">x=</script></span> existence at 2 yrs, <span class="math"><script type="math/tex">y=</script></span> existence at 20 yrs</li>
-<li><span class="math"><script type="math/tex">0=</script></span> dead, <span class="math"><script type="math/tex">1=</script></span> alive</li>
-</ul>
-</li>
-<li><span class="math"><script type="math/tex">x=1</script></span> (alive - assumed because you must exist in the first instance)</li>
-</ul>
-<p><strong>(case 1)</strong> if <span class="math"><script type="math/tex">x=1</script></span>, then <span class="math"><script type="math/tex">y=1</script></span> (assuming you don’t die between) <span class="math"><script type="math/tex">\therefore x = y</script></span><br>
-<strong>(case 2)</strong> if <span class="math"><script type="math/tex">y=1</script></span>, then <span class="math"><script type="math/tex">x=0</script></span> (because you killed yourself)<br>
-This is not a contradiction - it simply states that <span class="math"><script type="math/tex">x=0 \lor 1</script></span> (one <em>or</em> zero)</p>
-<p><strong><span class="math"><script type="math/tex">\therefore</script></span> both situations can exist (superposition) - suggesting multiple universe theory</strong></p>
-<ul>
-<li>Implies a link to quantum mechanics: dead or alive, or between (superposition):</li>
-<li><span class="math"><script type="math/tex">1 \over 2</script></span> probability of killing grandfather/self (two parallel realities - Murphy’s law)</li>
-</ul>
-<h3>Many worlds interpretation</h3>
-<ul>
-<li>different branches of superposition are different universes. Perception jumps between universes.</li>
-<li>time can be defined only by the physical state of things at the time - this is the only way it can be derived. therefore, by changing the physical state, you are creating a new "time (or timeline)</li>
-</ul></body>
-</html>
+++ /dev/null
-# Time
-
-## Presentism
-
-- The present time exists exclusively with its objects
-- There are no non-present objects
-- Past & future are human constructs
-- Memories etc are representations of the past in the present
-
-
-<pre>past present future
----------|----------
--->-->-->-->-->-->-></pre>
-
-## Eternalism
-
-- Fourth dimension
-- Distant times in past and future exist, like distant places
-- Past, present and future are equally real
-
-## Philosophers' Perspectives
-
-### Aristotle
-- we measure time as the "interval between two nows"
-- time $\propto$ change
-- subjective
-
-### Nietzsche
-- Time repeats infinitely in the exact same sequence
-- "Love your fate"
-
-### Augustine
-- Linking Aristotelian philosophy (infinitesimal yet eternal present) to God (always present)
-- Anything that lasts for a finite time comes from God, yet God is an eternal now
-- Also considered afterlife / multiple phases of time
-- Time tends towards non-existence, because the past and future are the only times that can really exist, and the present is infinitesimal (the only definite time).
-- Yet, time could not exist in infinitesimally small pieces, which go out of existence instantaneously.
-- Hence, time is a function of human perception and memory.
-
-#### Response
-Each premise of the argument seems to appear reasonable, but it is merely meta-analysis - building upon the Aristotelian idea that the present is infinitesimal. Also, there is no proof that time could not exist in infinitesimally small pieces - persistent memory does not disprove this. Events could span multiple "slices" of time. The conclusion that time is an exclusively human construct does not follow from the premise that the present could not exist infinitesimally.
-
-### Borges
-- We think time is a sequence of unique moments, but they can be repeatedf
-- Time = sensations e.g. reading Shakespeare makes you Shakespeare
-- Finite number of variables, therefore time can repeat itself
-
-### Einstein
-- We cannot measure time without being external to it
-- 1 s / s
+++ /dev/null
-# Circular forces
-
-## Velocity in a circle
-
-$$v={{2 \pi r} \over T}$$
-
-where $T$ is the period (time for one oscillation)
-
-(derived from $v={d \over t}$ where $d=2 \pi r = \pi D$)
-
-## Frequency and period
-
-$$ f={1 \over T}$$
-$$ t={1 \over f}$$
-
-## Centripetal acceleration & force
-
-$$a={{{v^2} \over r}={{4 \pi^2r}\over T^2}}$$
-
-where
-$a$ is centripetal acceleration
-$v$ is speed
-$r$ is radius
-$T$ is period
-
-and
-
-$v \perp a$
-
-We know that $F=ma$, so
-
-$$F={{mv^2}\over r}={{4\pi^2rm}\over T^2}$$
-
-## Banked track
-
-Forces acting:
-- Weight force $F_g$, vertically down
-- Normal (reaction) force $R$, perpendicular to slope
-
-
-- Net force $\Sigma F$ acts towards centre
-
-### Calculating the angle
-
-$$ tan \theta = {\Sigma F \over F_g} $$
-
-where
-$\Sigma F$ acts towards centre of circle
-$F_g$ is force by gravity on the moving object
-
-In terms of velocity..
-
-$$ \tan \theta = {v^2 \over rg}$$
-
-$$\therefore \theta = \tan^{-1}({v^2 \over rg})$$
-
-$$\therefore v=\sqrt{rg \tan \theta} $$
-
-- If $F_N > F_g$, passenger feels heavier
-- If $F_N < F_g$, passenger feels lighter
-
-### Minimum velocity in circle
-
-$$ v= \sqrt{gr}$$
-
-### Force and acceleration
-
-$$\Sigma F = F_N + mg$$
-
-## Vertical circular motion
-
-## Pulley-mass system
-
-$$a={{m_2g}\over{m_1+m_2}}$$
-
-where $m_2$ is the suspended mass
+++ /dev/null
-# Fields
-
-**Non-contact force**
-
-Types of fields:
-- gravitational
-- electric
-- magnetic
-
-Electric and magnetic fields are the two components of the electromagnetic field (effectively the same force). Permeates the universe.
-
-Field diagrams - N to S (current +ve to -ve)
-
-
-Field strength - inverse sq law
-
-## Gravitational fields
-
-### Field strength
-
-$$g={{F_g} \over m}$$
-
-## Electric fields
-
-### Field lines
-- +ve to -ve
-- start and end $\perp$ to surface
-- never cross
-
-### Field strength
-
-$$E = {F \over q}$$
-
-
-
-
-## Transformers
-
-Primary - input
-Secondary - output
-
-As $V \Rightarrow \infty, P_{loss} \Rightarrow 0$
-
-$$P_{loss} = I^2 R$$
-
-where $R$ is determined by resistivity per length of cable
-
-Therefore, power loss is proportional to the square of current.
-
-Electronmagnetism & electromagnetic induction
-
-Changing magnetic field generates current & vice versa
-
-Only for AC, because DC voltage is constant, so does not generate magnetic field
-
-Output is proportional to ratio primary : secondary
-
-
-$${V_{sec} \over V_{pri}} = {turns_{sec} \over turns_{pri}}$$
-
-Iron - good conductor of magnetic fields. Used as a core.
+++ /dev/null
-## Gravity
-
-Force that drives the universe - attractive force between masses
-1 of 4 fundamental forces
-
-$$
-{F _{g} }= {{Gm_1m_2}\over{r^2}}
-$$
-
-where
-$G=$ gravitational constant $=6.67 \times10^{-11}Nm^2kg^{-2}$
-$r=$ distance between *centres* of $m_1$ and $m_2$
-
-Area under force-distance graph = work = $\Delta E$
-
-# Newton
+++ /dev/null
-# Newtons' Laws
-
-> **Up is always positive**
-
-## First law
-
-Constant velocity if forces are balanced
-- object at rest remains at rest
-- object at motion remains in motion
-**only** if it does not experience an unbalanced force
-
-## Second law
-$$F_{net} = ma$$
-
-## Third law
-
-- Equal and opposite forces
-- Normal force (reaction) is in opposite direction to action force
-
-## Inclined plane
-
-$$F=ma=mg\sin\theta-{\mathit{friction}}$$
+++ /dev/null
-# Projectile motion
-
-- Horizontal velocity ($v_x$) remains constant if no drag
-
-- Vertical acceleration ($a_y$) $= -g$
-
-## Calculating velocity from $x$ and $y$ components
-
-${\displaystyle v={\sqrt {v_{x}^{2}+v_{y}^{2}\ }}}$
-
-## Displacement
-
-$x=v_{0}t\cos(\theta)$
-$y=v_{0}t\sin(\theta )-{\frac {1}{2}}gt^{2}$
-
-## Maximum height of an arc
-
-$h={\frac {u^{2}\sin ^{2}(\theta )}{2g}}$
-
-## Time of flight
-$y=u\ t\sin(\theta )-{\frac {1}{2}}gt^{2}$
+++ /dev/null
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- color: #56b6c2;
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- color: inherit;
-}
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- color: #abb2bf;
-}
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- color: #c678dd;
-}
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- color: #d19a66;
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- </head>
- <body class='markdown-preview'><h1>Special Relativity</h1>
-<h2>Postulates of special relativity</h2>
-<ol>
-<li>
-<p>Laws of physics are constant in all inertial reference frames</p>
-</li>
-<li>
-<p>Speed of light <span class="math"><script type="math/tex">c</script></span> is the same to all observers (proved by Michelson-Morley)</p>
-</li>
-</ol>
-<p><span class="math"><script type="math/tex">\therefore</script></span>, <span class="math"><script type="math/tex">t</script></span> must dilate as speed changes.</p>
-<p>Time and space - four dimensional relationship</p>
-<p><strong>Inertial reference frame</strong> - not accelerating</p>
-<p><strong>Proper time / length</strong> - measured by observer in same frame as events</p>
-<h2>Lorentz factor</h2>
-<span class="math"><script type="math/tex; mode=display">\gamma = {1 \over \sqrt{1-{v^2 \over c^2}}}
-</script></span>
-<p>Proper (time|length) denoted by (<span class="math"><script type="math/tex">t|l_0</script></span>) is time observed from moving frame.</p>
-<p>Applied:</p>
-<p><span class="math"><script type="math/tex">t = t_0 \gamma</script></span> (time - longer in moving frame)<br>
-<span class="math"><script type="math/tex">l={l_0 \over \gamma}</script></span> (length - only contracts in direction of motion - shorter in moving frame)<br>
-<span class="math"><script type="math/tex">m=m_0 \gamma</script></span> (mass dilation)</p>
-<span class="math"><script type="math/tex; mode=display">v=c \sqrt{1-{1 \over \gamma ^2}}
-</script></span>
-<p>(<span class="math"><script type="math/tex">c=3 \times 10^8</script></span> m / s)</p>
-<h2>Mass-energy equivalence</h2>
-<p><span class="math"><script type="math/tex">E_0=mc^2</script></span> (rest energy)</p>
-<p><span class="math"><script type="math/tex">E=\gamma mc^2</script></span> (total energy)</p>
-<ul>
-<li>mass is a dense form of energy</li>
-</ul>
-<h3>Relativistic momentum</h3>
-<span class="math"><script type="math/tex; mode=display">\rho = {mv \over {\sqrt{1-{v^2 \over c^2}}}} = \gamma mv=\gamma \rho_0
-</script></span>
-<p>Low speeds - little difference between relativistic & non-relativistic momentum<br>
-<span class="math"><script type="math/tex">\rho</script></span> approaches <span class="math"><script type="math/tex">\infty</script></span> as speed approaches <span class="math"><script type="math/tex">c</script></span><br>
-Impossible to reach speed of light (speed <span class="math"><script type="math/tex">c</script></span> requires <span class="math"><script type="math/tex">\infty</script></span> energy)<br>
-Photon has no mass - zero space or time</p>
-<h3>Total energy of an object</h3>
-<span class="math"><script type="math/tex; mode=display">E_{total}=E_k+E_{rest}=\rho mc^2
-</script></span>
-<h3>Kinetic energy</h3>
-<span class="math"><script type="math/tex; mode=display">E_k=(\gamma - 1)mc^2
-</script></span>
-<p>(takes relativity into account - use for fast objects)</p>
-<h3>Rest energy</h3>
-<span class="math"><script type="math/tex; mode=display">E_{rest}=mc^2
-</script></span>
-<p>(where <span class="math"><script type="math/tex">v<0.1c</script></span>) - this is the innate mass energy (proper energy)</p>
-<h3>Work</h3>
-<span class="math"><script type="math/tex; mode=display">W = \Delta E = \Delta mc^2
-</script></span>
-<h3>Velocity</h3>
-<span class="math"><script type="math/tex; mode=display">v={\rho \over {m \sqrt{1+{p^2 \over {m^2c^2}}}}}
-</script></span>
-<h3>Fusion / Fission equations</h3>
-<p>eV to J: multiply by <span class="math"><script type="math/tex">1.6\times 10^{-19}</script></span> (charge of an electron)</p>
-<h2>High altitude muons</h2>
-<ul>
-<li>Time dilation - more muons reach Earth than expected</li>
-<li>Normal half-life is <span class="math"><script type="math/tex">2.2 \mu s</script></span> in stationary frame of reference</li>
-<li>At close to <span class="math"><script type="math/tex">c</script></span>, muons observed from Earth have lifetime <span class="math"><script type="math/tex">> 2.2 \mu s</script></span></li>
-<li>Slower time - more time to travel, so more muons reach surface</li>
-</ul></body>
-</html>
+++ /dev/null
-# Special Relativity
-
-## Postulates of special relativity
-
-1. Laws of physics are constant in all inertial reference frames
-
-2. Speed of light $c$ is the same to all observers (proved by Michelson-Morley)
-
-$\therefore$, $t$ must dilate as speed changes.
-
-Time and space - four dimensional relationship
-
-**Inertial reference frame** - not accelerating
-
-**Proper time / length** - measured by observer in same frame as events
-
-## Lorentz factor
-
-$$\gamma = {1 \over \sqrt{1-{v^2 \over c^2}}}$$
-
-Proper (time|length) denoted by ($t|l_0$) is time observed from moving frame.
-
-Applied:
-
-$t = t_0 \gamma$ (time - longer in moving frame)
-$l={l_0 \over \gamma}$ (length - only contracts in direction of motion - shorter in moving frame)
-$m=m_0 \gamma$ (mass dilation)
-
-$$v=c \sqrt{1-{1 \over \gamma ^2}}$$
-
-($c=3 \times 10^8$ m / s)
-
-## Mass-energy equivalence
-
-$E_0=mc^2$ (rest energy)
-
-$E=\gamma mc^2$ (total energy)
-
-- mass is a dense form of energy
-
-### Relativistic momentum
-$$\rho = {mv \over {\sqrt{1-{v^2 \over c^2}}}} = \gamma mv=\gamma \rho_0$$
-
-Low speeds - little difference between relativistic & non-relativistic momentum
-$\rho$ approaches $\infty$ as speed approaches $c$
-Impossible to reach speed of light (speed $c$ requires $\infty$ energy)
-Photon has no mass - zero space or time
-
-### Total energy of an object
-
-$$E_{total}=E_k+E_{rest}=\rho mc^2$$
-
-### Kinetic energy
-
-$$E_k=(\gamma - 1)mc^2$$
-
-(takes relativity into account - use for fast objects)
-
-### Rest energy
-$$E_{rest}=mc^2$$
-(where $v<0.1c$) - this is the innate mass energy (proper energy)
-
-### Work
-
-$$W = \Delta E = \Delta mc^2$$
-
-### Velocity
-
-$$v={\rho \over {m \sqrt{1+{p^2 \over {m^2c^2}}}}}$$
-
-### Fusion / Fission equations
-
-eV to J: multiply by $1.6\times 10^{-19}$ (charge of an electron)
-
-## High altitude muons
-- Time dilation - more muons reach Earth than expected
-- Normal half-life is $2.2 \mu s$ in stationary frame of reference
-- At close to $c$, muons observed from Earth have lifetime $> 2.2 \mu s$
-- Slower time - more time to travel, so more muons reach surface
+++ /dev/null
-# Complex & Imaginary Numbers
-
-## Imaginary numbers
-
-$i^2 = -1$
-
-$\therefore i = \sqrt {-1}$
-
-### Simplifying negative surds
-
-$\sqrt{-2} = \sqrt{-1 \times 2}$
-
- $= \sqrt{2}i$
-
-
-## Complex numbers
-
-$\mathbb{C} = \{a+bi : a, b \in \mathbb{R} \}$
-
-General form: $z=a+bi$
-- $\operatorname{Re}(z) = a$
-- $\operatorname{Im}(z) = b$
-
-### Addition
-
-If $z_1 = a+bi$ and $z_2=c+di$, then
-
- $z_1+z_2 = (a+c)+(b+d)i$
-
-### Subtraction
-
-If $z_1=a+bi$ and $z_2=c+di$, then
-
- $z_1−z_2=(a−c)+(b−d)i$
-
-### Multiplication by a real constant
-
-If $z=a+bi$ and $k \in \mathbb{R}$, then
-
- $kz=ka+kbi$
-
-### Powers of $i$
-$i^0=1$
-$i^1=i$
-$i^2=-1$
-$i^3=-i$
-$i^4=1$
-$\dots$
-
-Therefore..
-- $i^{4n} = 1$
-- $i^{4n+1} = i$
-- $i^{4n+2} = -1$
-- $i^{4n+3} = -i$
-
-### Multiplying complex expressions
-
-If $z_1 = a+bi$ and $z_2=c+di$, then
- $z_1 \times z_2 = (ac-bd)+(ad+bc)i$
-
-### Conjugates
-
-If $z=a+bi$, conjugate of $z$ is $\overline{z} = a-bi$ (flipped operator)
-
-Also, $z \overline{z} = (a+bi)(a-bi) = a^2+b^2$
-
-- Multiplication and addition are associative
-
-### Modulus
-
-Distance from origin.
-$|{z}|=\sqrt{a^2+b^2}$
-
-$\therefore z \overline{z} = |z|^2$
-
-### Multiplicative inverse
-
-$z^{-1} = {1 \over z} = {{a-bi} \over {a^2+B^2}} = {\overline{z} \over {|z|^2}}$
-
-### Dividing complex numbers
-
-${{z_1} \over {z_2}} = {{z_1\ {z_2}^{-1}}} = {{z_1 \overline{z_2}} \over {{|z_2|}^2}}$
-
-(using multiplicative inverse)
-
-In practice, rationalise denominator:
-${z_1 \over z_2} = {{(a+bi)(c-di)} \over {c^2+d^2}}$
-
-## Argand planes
-
-- Geometric representation of $\mathbb{C}$
-- Horizontal $= \operatorname{Re}(z)$; vertical $= \operatorname{Im}(z)$
-- Multiplication by $i$ results in an anticlockwise rotation of $\pi \over 2$
-
-## Solving complex quadratics
-
-To solve $z^2+a^2=0$ (sum of two squares):
-
-$z^2+a^2=z^2-(ai)^2$
- $=(z+ai)(z-ai)$
-
-## Polar form
-
-General form:
-$z=r \operatorname{cis} \theta$
-$= r\operatorname{cos}\theta+r\operatorname{sin}\theta i$
-
-where
-- $z=a+bi$
-- $r$ is the distance from origin, given by Pythagoras ($r=\sqrt{x^2+y^2}$)
-- $\theta$ is the argument of $z$, CCW from origin
-
-Note each complex number has multiple polar representations:
-$z=r \operatorname{cis} \theta = r \operatorname{cis} (\theta+2 n\pi$) where $n$ is integer number of revolutions
-
-### Multiplication and division in polar form
-
-$z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)$ (multiply moduli, add angles)
-
-${z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)$ (divide moduli, subtract angles)
-
-## de Moivres' Theorum
-
-$(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta)$
+++ /dev/null
-<h1 id="complex-imaginary-numbers">Complex & Imaginary Numbers</h1>
-<h2 id="imaginary-numbers">Imaginary numbers</h2>
-<p><span class="math inline"><em>i</em><sup>2</sup> = −1</span></p>
-<p><span class="math inline">$\therefore i = \sqrt {-1}$</span></p>
-<h3 id="simplifying-negative-surds">Simplifying negative surds</h3>
-<p><span class="math inline">$\sqrt{-2} = \sqrt{-1 \times 2}$</span></p>
-<p> <span class="math inline">$= \sqrt{2}i$</span></p>
-<h2 id="complex-numbers">Complex numbers</h2>
-<p><span class="math inline">ℂ = {<em>a</em> + <em>b</em><em>i</em> : <em>a</em>, <em>b</em> ∈ ℝ}</span></p>
-<p>General form: <span class="math inline"><em>z</em> = <em>a</em> + <em>b</em><em>i</em></span> - <span class="math inline">Re(<em>z</em>)=<em>a</em></span> - <span class="math inline">Im(<em>z</em>)=<em>b</em></span></p>
-<h3 id="addition">Addition</h3>
-<p>If <span class="math inline"><em>z</em><sub>1</sub> = <em>a</em> + <em>b</em><em>i</em></span> and <span class="math inline"><em>z</em><sub>2</sub> = <em>c</em> + <em>d</em><em>i</em></span>, then</p>
-<p> <span class="math inline"><em>z</em><sub>1</sub> + <em>z</em><sub>2</sub> = (<em>a</em> + <em>c</em>)+(<em>b</em> + <em>d</em>)<em>i</em></span></p>
-<h3 id="subtraction">Subtraction</h3>
-<p>If <span class="math inline"><em>z</em><sub>1</sub> = <em>a</em> + <em>b</em><em>i</em></span> and <span class="math inline"><em>z</em><sub>2</sub> = <em>c</em> + <em>d</em><em>i</em></span>, then</p>
-<p> <span class="math inline"><em>z</em><sub>1</sub> − <em>z</em><sub>2</sub> = (<em>a</em> − <em>c</em>)+(<em>b</em> − <em>d</em>)<em>i</em></span></p>
-<h3 id="multiplication-by-a-real-constant">Multiplication by a real constant</h3>
-<p>If <span class="math inline"><em>z</em> = <em>a</em> + <em>b</em><em>i</em></span> and <span class="math inline"><em>k</em> ∈ ℝ</span>, then</p>
-<p> <span class="math inline"><em>k</em><em>z</em> = <em>k</em><em>a</em> + <em>k</em><em>b</em><em>i</em></span></p>
-<h3 id="powers-of-i">Powers of <span class="math inline"><em>i</em></span></h3>
-<p><span class="math inline"><em>i</em><sup>0</sup> = 1</span> <span class="math inline"><em>i</em><sup>1</sup> = <em>i</em></span> <span class="math inline"><em>i</em><sup>2</sup> = −1</span> <span class="math inline"><em>i</em><sup>3</sup> = −<em>i</em></span> <span class="math inline"><em>i</em><sup>4</sup> = 1</span> <span class="math inline">…</span></p>
-<p>Therefore.. - <span class="math inline"><em>i</em><sup>4<em>n</em></sup> = 1</span> - <span class="math inline"><em>i</em><sup>4<em>n</em> + 1</sup> = <em>i</em></span> - <span class="math inline"><em>i</em><sup>4<em>n</em> + 2</sup> = −1</span> - <span class="math inline"><em>i</em><sup>4<em>n</em> + 3</sup> = −<em>i</em></span></p>
-<h3 id="multiplying-complex-expressions">Multiplying complex expressions</h3>
-<p>If <span class="math inline"><em>z</em><sub>1</sub> = <em>a</em> + <em>b</em><em>i</em></span> and <span class="math inline"><em>z</em><sub>2</sub> = <em>c</em> + <em>d</em><em>i</em></span>, then <span class="math inline"><em>z</em><sub>1</sub> × <em>z</em><sub>2</sub> = (<em>a</em><em>c</em> − <em>b</em><em>d</em>)+(<em>a</em><em>d</em> + <em>b</em><em>c</em>)<em>i</em></span></p>
-<h3 id="conjugates">Conjugates</h3>
-<p>If <span class="math inline"><em>z</em> = <em>a</em> + <em>b</em><em>i</em></span>, conjugate of <span class="math inline"><em>z</em></span> is <span class="math inline">$\overline{z} = a-bi$</span> (flipped operator)</p>
-<p>Also, <span class="math inline">$z \overline{z} = (a+bi)(a-bi) = a^2+b^2$</span></p>
-<ul>
-<li>Multiplication and addition are associative</li>
-</ul>
-<h3 id="modulus">Modulus</h3>
-<p>Distance from origin. <span class="math inline">$|{z}|=\sqrt{a^2+b^2}$</span></p>
-<p><span class="math inline">$\therefore z \overline{z} = |z|^2$</span></p>
-<h3 id="multiplicative-inverse">Multiplicative inverse</h3>
-<p><span class="math inline">$z^{-1} = {1 \over z} = {{a-bi} \over {a^2+B^2}} = {\overline{z} \over {|z|^2}}$</span></p>
-<h3 id="dividing-complex-numbers">Dividing complex numbers</h3>
-<p><span class="math inline">${{z_1} \over {z_2}} = {{z_1\ {z_2}^{-1}}} = {{z_1 \overline{z_2}} \over {{|z_2|}^2}}$</span></p>
-<p>(using multiplicative inverse)</p>
-<p>In practice, rationalise denominator: <span class="math inline">${z_1 \over z_2} = {{(a+bi)(c-di)} \over {c^2+d^2}}$</span></p>
-<h2 id="argand-planes">Argand planes</h2>
-<ul>
-<li>Geometric representation of <span class="math inline">ℂ</span></li>
-<li>Horizontal <span class="math inline">=Re(<em>z</em>)</span>; vertical <span class="math inline">=Im(<em>z</em>)</span></li>
-<li>Multiplication by <span class="math inline"><em>i</em></span> results in an anticlockwise rotation of <span class="math inline">$\pi \over 2$</span></li>
-</ul>
-<h2 id="solving-complex-quadratics">Solving complex quadratics</h2>
-<p>To solve <span class="math inline"><em>z</em><sup>2</sup> + <em>a</em><sup>2</sup> = 0</span> (sum of two squares):</p>
-<p><span class="math inline"><em>z</em><sup>2</sup> + <em>a</em><sup>2</sup> = <em>z</em><sup>2</sup> − (<em>a</em><em>i</em>)<sup>2</sup></span> <span class="math inline">=(<em>z</em> + <em>a</em><em>i</em>)(<em>z</em> − <em>a</em><em>i</em>)</span></p>
-<h2 id="polar-form">Polar form</h2>
-<p>General form: <span class="math inline"><em>z</em> = <em>r</em>cis<em>θ</em></span> <span class="math inline">=<em>r</em>cos<em>θ</em> + <em>r</em>sin<em>θ</em><em>i</em></span></p>
-<p>where - <span class="math inline"><em>z</em> = <em>a</em> + <em>b</em><em>i</em></span> - <span class="math inline"><em>r</em></span> is the distance from origin, given by Pythagoras (<span class="math inline">$r=\sqrt{x^2+y^2}$</span>) - <span class="math inline"><em>θ</em></span> is the argument of <span class="math inline"><em>z</em></span>, CCW from origin</p>
-<p>Note each complex number has multiple polar representations: <span class="math inline"><em>z</em> = <em>r</em>cis<em>θ</em> = <em>r</em>cis(<em>θ</em> + 2<em>n</em><em>π</em></span>) where <span class="math inline"><em>n</em></span> is integer number of revolutions</p>
-<h3 id="multiplication-and-division-in-polar-form">Multiplication and division in polar form</h3>
-<p><span class="math inline"><em>z</em><sub>1</sub><em>z</em><sub>2</sub> = <em>r</em><sub>1</sub><em>r</em><sub>2</sub>cis(<em>θ</em><sub>1</sub> + <em>θ</em><sub>2</sub>)</span> (multiply moduli, add angles)</p>
-<p><span class="math inline">${z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)$</span> (divide moduli, subtract angles)</p>
-<h2 id="de-moivres-theorum">de Moivres' Theorum</h2>
-<p><span class="math inline">(<em>r</em>cis<em>θ</em>)<sup><em>n</em></sup> = <em>r</em><sup><em>n</em></sup>cis(<em>n</em><em>θ</em>)</span></p>
+++ /dev/null
-# Sequences and Series
-
-$\{\ a,\ b,\ c,\ d\ ...\ \}$
-
-
-- $n^{th}$ iteration: $t_n$
-- $n$ can start at $0$ or $1$
-
-## Defining sequences
-- sequence rule in terms of $n$ e.g. $t_n = 2n$
-
-- recurrence relation - value of term is derived from previous term (recursion)
-e.g. $\ t_n=t_{n-1}+7,\ t_1=4$
- - $t_1=4$
- - $t_2=4+7=11$
- - $t_3=11+7=17$
- - $\dots$
-
-## Arithmetic sequences
-
-$$ t_n = a+(n-1)d$$
-
-where
-$a=$ first term
-$d=$ common difference
-
-### Arithmetic mean
-
-$$m_a = {{a+b} \over 2}$$
-
-### Arithmetic series
-
-$\sum$ of all terms in an arithmetic sequence.
-
-e.g. A sequence is defined by $t_n = a+(n-1)d$
-
-$s_n = {n \over 2}(2a+(n-1)d)$ <--sum of $n$ terms, including $t_n$ (works only with $t_1$)
-where
-- $a=t_1$ (first term)
-- $l=t_n$ (last term)
-
-## Geometric sequences
-
-$$t_n=ar^{n-1}$$
-where
-$a =$ first term
-$r =$ common ratio of successive terms ($r={t_k \over t_{k-1}}$)
-
-### Geometric mean
-
-$m_g$ of $a$ and $b$ is $\sqrt{ab}$
-
-If $a, c, b$ are positive and consecutive terms in a geometric sequence, then:
-${c \over a } = {b \over c} \therefore c = \sqrt{ab}$,
-
-### Geometric series
-
-$\sum$ of all terms in a geometric sequence.
-
-e.g. $s_n = a + ar + ar^2 + ar^2 + \dots + ar^{n-1}$
-
-$rs_n = ar+ar^2+ar^3+ar^4+\dots+ar^{n}$
-
-$rs_n - s_n = -a + ar^n$
-
----
-$s_n = {{a(r^n-1)} \over {r-1}}$
-or
-$s_n = {{a(r^n-1)}\over r-1}$
-
-## Infinite series
-
-If $-1 < r -1$, the infiniteg eometric series $a+ar+ar^2 \dots$ is convergent.
-
-Sum to infinity is given by
-
-$$s_\infty={a \over {1-r}}$$
-
-Tennis ball question - remember down **and** up strokes. Multiply down strokes by 2, subtract 1.
+++ /dev/null
-# Trigonometry
-
-## Unit circle
-
-For each angle, there is a corresponding point on the unit circle ($r=1$).
-
-- $\cos \theta$ is the $x$-coordinate of this point
-- $\sin \theta$ is the $y$-coordinate
-
-
-
-## Trigonometric ratios
-
-
-## Cosine rule