eng: central assertion definition

Reviewed-on: https://git.eggworld.tk/eggy/eifueo/pulls/7

docs/eng3uz.md

@@ -2,7 +2,7 @@
 
 The course code for this page is **ENG3UZ**.
 
-## Literary Techniques/Devices
+## Literary techniques/devices
 
 ### Description
 
@@ -134,6 +134,20 @@ The course code for this page is **ENG3UZ**.
  - Theme: The "main idea" or underlying meaning of a literary work, which can be given directly or indirectly.
 	- e.g., *"Never forget that* you are royalty, *and that hundreds of thousands of souls have suffered and perished so you could become what you are. By their sacrifices, you have been given the comforts you take for granted. Always remember them, so that their sacrifices shall never be without meaning."* (*Eon Fable*, ScytheRider)
 
+## Central assertion
+
+ - "Purpose" of the poem
+
+## Commentary
+
+ - Present tense all the time
+ - Use numbers in parentheses for sources
+ - Quotes everywhere
+
+ - Topic sentence 
+ - Introduce what was used by Neruda
+ - Evidence of where they were used and why/how they were used 
+
 ## Resources
 
  - [Analysis of a Poem](/resources/g11/central-asserion-1.pdf)

docs/mhf4u7.md

@@ -55,7 +55,7 @@ A frequency distribution table includes:
 
  - A number of classes, all of the same width.
 	- This number is arbitrarily chosen, but a commonly used formula is $\lceil\sqrt{\text{# of elements}}\rceil$.
-	- The width (size) of each class is $\lceil\frac{\text{max value} - \text{min value}}{\text{number of classes}}\rceil$.
+	- The width (size) of each class is $\lceil\frac{\text{max} - \text{min}}{\text{# of classes}}\rceil$.
 	- Each class includes its lower bound and excludes its upper bound ($\text{lower} ≤ x < \text{upper}$)
 	- The **relative frequency** of a data set is the percentage of the whole data set present in that class in decimal form.
  - The number of values that fall under each class.
@@ -97,10 +97,9 @@ The first and third quartiles are the median of the **[minimum, median)** and **
     When working with grouped data given in ranges, the actual data is unavailable. The five numbers above are instead:
 
     - The minimum value is now the lower class boundary of the lowest class.
-    - The first quartile is the midpoint of the class it resides in.
-    - The median is the midpoint of the class it resides in.
-    - The third quartile is the midpoint of the class it resides in.
+    - The first and third quartiles, as well as the median, are now found by guesstimating the value on a cumulative frequency curve.
     - The maximum value is now the upper class boundary of the highest class. If the highest value is excluded (e.g., $90≤x<100$), it also must be excluded when representing data (e.g., open dot instead of filled dot).
+    - A specific percentile can be found by guesstimating the value on a cumulative frequency curve.
 
 The **interquartile range (IQR)** is equal to $Q_3 - Q_1$ and represents the range where 50% of the data lies.
 
@@ -150,9 +149,123 @@ A **box-and-whisker plot** is a visual representation of the **"5-number summary
 
 ### Measures of central tendency
 
+The **mean** is the sum of all values divided by the total number of values. $\bar{x}$ represents the mean of a sample while $µ$ represents the mean of a population.
+
+$$\bar{x}=\frac{\sum x}{n}$$ where $n$ is equal to the number of values in the data set.
+
+In grouped data, the mean can only be estimated, and is equal to the average of the sum of midpoint of all classes multiplied by their class frequency.
+
+$$\bar{x} = \frac{\sum x_i f_i}{n}$$ where $x_i$ is the midpoint of the $i$th class and $f_i$ is the frequency of the $i$th class.
+
+The **median** is the middle value when the data set is sorted. If the data set has an even number of values, the median is the mean of the two centre-most values.
+
+In grouped data, the median class is the class of the $\frac{n+1}{2}$th value if the number of values in the class is odd or the $\frac{n}{2}$th value otherwise.
+
+The **mode** is the value that appears most often.
+
+!!! definition
+    - **Unimodal**: A data set with one mode.
+    - **Bimodal**: A data set with two modes.
+    - **Multimodal**: A data set with more than two modes.
+    - **No mode**: A data set with no values occurring more than once.
+
+In grouped data, the **modal class** is the class with the greatest frequency.
+
+### Measures of dispersion
+
+These are used to quantify the variability or spread of the data set.
+
+The **range** of a data set is simple to calculate but is easily thrown off by outliers.
+
+$$R = \max - \min$$
+
+The **variance** ($\sigma^2$) and **standard deviation** ($\sigma$) of a data set are more useful. The standard deviation indicates how closely the values of a data set are clustered around the mean.
+
+$$\sigma = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{n}}$$ where $f_i$ is the frequency of the $i$th class, $x_i$ is the midpoint of the $i$th class, $\bar{x}$ is the mean of the whole data set, and $n$ is the number of values in the data set.
+
+For ungrouped data, assume $f_i = 1$.
+
+In a typical bell-shaped distribution:
+
+ - 68% of data lie within 1 standard deviation of the mean ($\bar{x} ± \sigma$)
+ - 95% of data lie within 2 standard deviations of the mean ($\bar{x} ± 2\sigma$)
+ - 99.7% of data lie within 3 standard deviations of the mean ($\bar{x} ± 3\sigma$)
+ - any data outside 3 standard deviations of the mean can be considered outliers
+
+!!! info
+    The **points of inflection** (when the curve changes direction) of a normal bell curve occur at $\bar{x} ± \sigma$.
+
+### Data transformation
+
+When performing an operation with a constant value to a whole data set:
+
+| Operation | Effect on mean | Effect on standard deviation |
+| --- | --- | --- |
+| Addition/subtraction | Increased/decreased by constant | No change |
+| Multiplication/division | Multiplied/divided by constant | Multiplied/divided by constant |
+
+
+### Linear correlation and regression
+
+!!! definition
+    - **Interpolation**: The prediction of values within the range of a data set.
+    - **Extrapolation**: The prediction of values outside the range of a data set. This tends to be less reliable than interpolation as it is unknown if the model is accurate outside of the range of the data set..
+
+A scatter plot is used to help find trends and relationships between variables, which is primarily used to predict results not in the data set.
+
+If there is a clear trend in the data, there is said to be a **correlation** between the independent and dependent variables.
+
+ - If the line has an upward trend, it has a positive correlation.
+ - If the line has a downward trend, it has a negative correlation.
+
+The strength of the correlation ranges from none, weak, moderate, strong, and perfect, where the latter shows a line passing through all data points.
+
+The line of best fit may not be linear. It may be quadratic, exponential, logarithmic, or there might not be a line of best fit at all. In the latter case, there is **no correlation**.
+
+**Correlation does not imply causation**. There may be an external **confounding factor** which causes both trends, instead.
+
+!!! example
+    If ice cream consumption increases as deaths from drowning increase, it does not mean that drowning causes people to eat more ice cream. The confounding factor of summer increases ice cream consumption and frequency of swimming, which leads to more people drowning.
+
+To find the **regression line** (line of best fit), a mean data point is required. The mean data point is a new point located at the mean of all x- and y-coordinates, or $M = (\bar{x}, \bar{y})$. The regression line then is the line that passes through the mean point while minimising the *vertical* distance from every data point. This is most easily performed on a graphing display calculator (GDC), but can be calculated manually if needed.
+
+The **least squares regression** is used to find the equation of a line that passes through the mean point for which the *square* of the vertical distance between the line and all data points (the residuals) is minimised for each point. It involves forming a line such that the sum of all residuals is $0$, and the sum of all residuals squared is minimised.
+
+Alternatively, to manually guesstimate a linear line of best fit, a line can be drawn from the mean point to a point that best appears to lie on the line of best fit.
+
+The **Pearson product-moment correlation coefficient** (more commonly known as *Pearson's $r$* or the *$r$-value*) quantifies the **correlation strength** of a line of best fit, or how well the line of best fit fits. This value is such that $-1≤r≤1$, where
+
+ - $r>0$ is a positive correlation
+ - $r<0$ is a negative correlation
+ - $|r|=1$ is a perfect correlation
+ - $0.7≤|r|<1$ is a strong correlation
+ - $0.3≤|r|<0.7$ is a weak to moderate correlation
+ - $0≤|r|<0.3$ is no correlation, so that no line of best fit can be drawn.
+
+## 5 - Calculus
+
+### Rate of change
+
+The **average rate of change (ARoC)** between points $P(a, f(a))$ and $Q(a + h, f(a+h))$ is represented by the slope of the **secant line ($m_s$)**. Therefore, as slope is the difference in rise over the difference of run ($\frac{\Delta y}{\Delta x}$), the slope of the secant line can be expressed as
+$$m_s = \frac{f(a+h)-f(a)}{h}, h ≠ 0$$
+
+This is known as the **difference quotient**.
+
+The **instantaneous rate of change (IRoC)** at point $P(a, f(a))$ is represented by the slope of the **tangent line ($m_T$)**. The slope of the tangent line can be found by finding the difference quotient with $h$ as a very small value, e.g., $0.001$.
+
+### Sequences
+
+A sequence is a **function** with a domain of all positive integers in sequence, but uses subscript notation ($t_n$) instead of function notation ($f(x)$).
+
+### Limits
+
+### Evaluating limits
+
 ## Resources
 
  - [IB Math Analysis and Approaches Syllabus](/resources/g11/ib-math-syllabus.pdf)
  - [IB Math Analysis and Approaches Formula Booklet](/resources/g11/ib-math-data-booklet.pdf)
  - [Calculus and Vectors 12 Textbook](/resources/g11/calculus-vectors-textbook.pdf)
  - [Course Pack Unit 1: Descriptive Statistics](/resources/g11/s1cp1.pdf)
+ - [Course Pack Unit 2: Limits and Rate of Change](/resources/g11/s1cp2.pdf)
+ - [TI-84 Plus Basic Calculator Functions](/resources/g11/ti-84-plus.pdf)

docs/sch3uz.md

@@ -2,17 +2,228 @@
 
 The course code for this page is **SCH3UZ**.
 
+## Designing a scientific investigation
+
+### Scope
+
+The scope of an experiment goes at the very beginning of it. It includes a general introduction to the topic of investigation as well as personal interest.
+
+### Research question
+
+The research question of an experiment is a hyper-focused and specific question related to the topic. It contains and asks about the effect of an **independent variable** on a **dependent variable**.
+
+### Background information and hypothesis
+
+!!! note
+    This section can instead be placed immediately before the research question depending on the experiment.
+
+In this section, scientific theories are provided to help the reader understand the rationale of the question, the design of the experiment, and data processing measures. If any theoretical/literature values are used, they should be introduced here.
+
+A hypothesis consists of a justified prediction of the expected outcome and should be integrated with any background information.
+
+### Variables
+
+!!! definition
+    - The **independent** variable is the variable that is explicitly changed to attempt to affect the dependent variable.
+    - The **dependent** variable is the variable that is directly monitored and measured in the experiment and is expected to change if the independent variable changes.
+    - **Controlled** variables (also known as "control variables") are variables that should be kept constant so they do not affect the dependent variable.
+
+The independent variable, dependent variable, and any controlled variables should be listed under this section.
+
+### Materials
+
+A list of materials and equipment should be listed here, as well as their precision. If a controlled variable needs to be measured, any instruments that would be used to do so should also be listed here.
+
+### Procedure
+
+A clear, detailed, and concise set of instructions written in *past tense* should be placed in this section as either a numbered list or descriptive paragraph. To reduce confusion, if a numbered list is used, referring directly to numbers should be avoided, and referring to numbers recursively must *never* happen. A procedure must include:
+
+ - a clear, titled, labelled, and annotated diagram
+ - instructions for recording data (including for controlled variables)
+
+If necessary, a "setup" section can be added as preparatory steps should not be listed in the main procedure.
+
+### Data collection
+
+Data should be collected in an organised and titled table that should be prepared before the experiment. To reduce plagiarism, the data table must be verified by a teacher before leaving the lab space. After verification, **no new data** can be added. During an experiment that spans multiple days, this data must be verified every day. The data table must include:
+
+ - units with uncertainty, typically in the table header
+ - *qualitative* data (quantitative data can be optional in some experiments)
+ - repeated data/controlled variables, typically in the title
+
+Only **raw data** prior to any processing or calculations, with the exception of averages, should be present in the data table.
+
+A data table should be as concise as possible, and redundancy should be minimised. In that vein, trial numbers should *not* be recorded unless that data is relevant.
+
+!!! example
+    **Effect of Fat Content on Sugar Content in Ice Cream**
+    
+    | Fat Content (g ± 0.1 g) | Sugar Content (g ± 0.1 g) | Notes |
+    | --- | --- | --- |
+    | 2.0 | 5.1 | - strawberry ice cream |
+    | 0.1 | 2.3 | - mint chocolate chip ice cream |
+
+### Data processing
+
+A single sample calculation showing all steps should be present and clearly explained. The rest of the data can be processed without describing any steps. A **single** graph may be included if needed.
+
+Some general rules include:
+
+ - units and uncertainties must be present in all calculations
+ - simple operations such as averages and conversions (e.g., g to kg) do not need to be explained
+ - the graph, if any, should span a full page and should directly answer the research question
+
+A final, reorganised, and processed data table should be present here, showing only relevant information.
+
+### Conclusion and evaluation
+
+This section should be free of any new background information or calculations. It should, in sequence:
+
+ - summarise the results of the experiment without connecting it to the hypothesis
+ - identify whether the results of the experiment agree or disagree with the hypothesis
+ - evaluate 3–5 systematic errors (usually) present in the experiment, both in the procedure and in data collection/processing, in **decreasing** order of impact to the experiment
+
+The evaluation of systematic errors should include:
+
+ - a description of the error
+ - how the error affected the data
+ - how the error affected the final result
+ - how the error can be remedied with available school resources
 ## 11.1 - Uncertainties and errors in measurement and results
 
-!!! info
-    Please see [SL Physics](/sph3u7/#12-uncertainties-and-errors) for more information.
+Please see [SL Physics#Uncertainties and errors](/sph3u7/#12-uncertainties-and-errors) for more information.
 
 ## 11.2 - Graphical techniques
 
+When plotting a graph:
 
+ - plot the independent variable on the horizontal axis and the dependent variable on the vertical axis
+ - label the axes, ensuring that the labels include units
+ - choose an appropriate scale for each axis
+ - give the graph an appropriate title
+ - draw a line of best fit
+
+### Titles
+
+The title of a graph should clearly indicate what the graph represents under what conditions in **title case**, so that any onlooker should be able to identify the experiment. It should not include "vs." Any legends present should be located under the graph.
+
+??? example
+    "Effect of Cat Deaths on Suicides in New Zealand"
+
+### Error bars
+
+Please see [SL Physics#Error bars](/sph3u7/#error-bars) for more information.
+
+### Line of best fit
+
+Please see [SL Physics#Uncertainty of gradient and intercepts](/sph3u7/#uncertainty-of-gradient-and-intercepts) for more information.
 
 ## 11.3 - Spectroscopic identification of organic compounds
 
+## 12 - Atomic structure
+
+!!! definition
+    - The **effective nuclear charge** ($Z_\text{eff}$) is the net positive charge (attraction to the nucleus) experienced by an electron in an atom.
+    - **Electron shielding** describes the decrease in the effective nuclear charge of an electron because of the repulsion of other electrons in lower-energy shells.
+
+**Atomic notation** is used to represent individual atoms or ions. It is written in the form $^M_Z \text{Symbol}^\text{Charge}$, where $M$ is the mass number of the particle and $Z$ is the atomic number of the particle.
+
+!!! example
+    - $^1_1 \text{H}^{+}$ is the atomic notation for the most common hydrogen ion.
+    - $^{16}_8 \text{O}^{2-}$ is the atomic notation for the most common oxygen ion.
+    - $^{20}_{10} \text{Ne}$ is the atomic notation for the most common neon atom.
+
+### Isotopes
+
+Isotopes are atoms of the same element but with different masses, or alternatively, atoms with the same number of protons but with different numbers of neutrons.
+
+**Radioisotopes** are isotopes that are unstable (will spontaneously decay, are radioactive). Unstable atoms **decay** (break down) into one or more different isotopes of a different element. The **half-life** of a radioisotope is the time it takes for 50% of a sample's atoms to decay.
+
+!!! warning
+    Radioisotopes are dangerous! They emit radiation, which is not at all good for human health in the vast majority of cases. However, there are also useful applications for radioisotopes today. For example, Cobalt-60 is used in radiation therapy to kill cancer tumours by damaging their DNA.
+
+### Atomic mass
+
+The mass of every atom is represented relative to 1/12th of a carbon-12 atom. This mass is either unitless or expressed in terms of **atomic mass units (amu or u)**. On the periodic table, the **relative atomic mass** ($A_r$) is shown, which is the sum of the masses of each isotope combined with their natural abundance (%abundance).
+
+$$A_r = \text{%abundance}×\text{mass number of isotope}$$
+$$m_a = \Sigma A_r$$
+
+When calculating the atomic mass from the graph from a **mass spectrometer**, the sum of the natural abundances of each isotope may not equal 100 or 1 (not in %abundance). In this case, calculation of %abundance will need to be done before solving for $m_a$.
+
+A mass spectrometer may also provide mass in the form of $M/Z$, which represents mass over charge. For the sake of simplicity, $Z=1$, so $M/Z$ represents the mass of a particle.
+
+### Atomic radius
+
+The atomic radius of an atom is the distance from the centre of the nucleus to approximately the outer boundary of the electron shell. This cannot be directly measured by scientists.
+
+### Ionisation energy
+
+The first ionisation energy of an element is the minimum amount of energy required to remove one mole of electrons from one mole of *gaseous* atoms to form a mole of gaseous ions, so that
+$$\text{Q}_\text{(g)} \rightarrow \text{Q}_\text{(g)}^+ + \text{e}^-$$
+
+Any subsequent ionisation energies of an element are the minimum amount of energy required to remove one *additional* mole of electrons. For example, the second ionisation energy would involve
+$$\text{Q}_\text{(g)}^+ \rightarrow \text{Q}_\text{(g)}^{2+} + \text{e}^-$$
+
+It requires vastly more energy to remove an electron from a filled valence shell compared to an unfilled valence shell.
+
+### Electron affinity
+
+The electron affinity of an atom is the amount of energy **required** or **released** to *add* an electron to a neutral *gaseous* atom to form a negative ion, such that
+$$\text{Q}_\text{(g)} + \text{e}^- \rightarrow \text{Q}^-_\text{(g)}$$ 
+
+If energy is released, the atom has a **negative** electron affinity, and will form a stable ion.
+
+If energy is required, the atom has a **positive** electron affinity, and will form an unstable ion (the ion will spontaneously decay).
+
+### Electronegativity
+
+The electronegativity of an atom represents the ability of that atom to attract a pair of electrons in a **covalent bond**. It ranges from $0$ to $4$ on the Pauling scale. As electronegativity increases, the atom more strongly holds on to the electrons in its covalent bond, so the pair of electrons in that bond spend more time around the atom with the higher electronegativity.
+
+### Reactivity
+
+The reactivity of an element is how "willing" it is to give up or gain electrons to fill its valence shell.
+
+The reaction of an **alkali metal** with water always forms a hydroxide and hydrogen gas. For example, lithium reacts with water such that:
+$$2\text{Li}_\text{(s)} + 2\text{H}_2\text{O}_\text{(l)} \rightarrow 2\text{LiOH}_\text{(aq)} + \text{H}_{2 (g)}$$
+
+The reaction of a **halogen** with hydrogen gas always forms a hydride. For example, fluorine reacts with hydrogen gas such that:
+$$\text{Fl}_\text{(g)} + \text{H}_\text{2 (g)} \rightarrow 2\text{HFl}_\text{(g)}$$
+
+###Models
+
+Please see [SL Physics#Models](/sph3u7/#models) for more information.
+
+### Periodic trends
+
+Some trends in the periodic table include:
+
+ - atomic radius decreases when going across a period and increases when going down a group
+ - ionic radius decreases when going across a period for groups 1–13, then sharply increases and then increases for groups 15–17; it increases when going down a group
+ - electron affinity increases when going across a period and decreases when going down a group
+ - ionisation energy increases when going across a period and decreases when going down a group
+ - electronegativity increases when going across a period and decreases when going down a group
+ - reactivity of alkali metals increases when going down the group
+ - reactivity of halogens decreases when going down the group
+
+When explaining these trends in the periodic table, it is best to use the following basic concepts to build on to larger points.
+
+Across a period, the number of shells occupied by the electrons is the same but the number of protons in the nucleus increases. Therefore,
+
+ - the attraction of each electron to the nucleus (effective nuclear charge) increases as the number of protons increases
+ - shielding is unchanged as the number of electrons between the valence electrons and the nucleus is the same
+
+Down a period, the number of shells occupied by the electrons increases, so valence electrons are further from the nucleus. Therefore,
+
+ - the attraction of valence electrons to the nucleus decreases due to the increasing distance
+ - shielding increases due to the increasing number of electrons between the valence electrons and the nucleus
+
+!!! example
+    To explain why there is a trend of decreasing atomic radius across a period:
+
+    - As the number of protons and electrons increase together, but the number of electron shells does not change, the effective nuclear charge of each electron increases, while the effect of shielding remains unchanged.
+    - This increased effective nuclear charge reduces the atomic radius compared to other atoms before it.
+
 ## Resources
 
  - [IB Chemistry Data Booklet](/resources/g11/ib-chemistry-data-booklet.pdf)

docs/sph3u7.md

@@ -123,18 +123,23 @@ See [Dealing with Uncertainties](/resources/g11/physics-uncertainties.pdf) for h
 
 ### Error bars
 
-Error bars represent the uncertainty of the data, typically representing that data point's standard deviation, and can be both horizontal or vertical. A data point with uncertain values is written as $(x ± \Delta x, y ± \Delta y)$
+Error bars represent the uncertainty of the data and can be both horizontal or vertical. They are almost always required for both the independent and dependent variables. A data point with uncertain values is written as $(x ± \Delta x, y ± \Delta y)$
 
 <img src="/resources/images/error-bars.png" width=600>(Source: Kognity)</img>
 
+If the error bars of a data point are too small to see, note at the bottom of the graph that error bars are too small to see.
+
 ### Uncertainty of gradient and intercepts
 
 !!! note "Definition"
     - The **line of best fit** is the line that passes through **as many error bars as possible** while passing as closely as possible to all data points.
-    - The **minimum and maximum lines** are lines that minimise/maximise their slopes while passing through the first and last **error bars**.
+    - The **minimum and maximum lines** are lines that minimise/maximise their slopes while passing through as many **error bars** as possible.
 
 !!! warning
     - Use solid lines for lines representing **continuous data** and dotted lines for **discrete data**.
+    - The line of best fit may not be a straight line.
+    - Be wary and verify the results of a best fit line from software, as outliers and data trends may not be recognised by a computer.
+    - It is better to leave a data point in the graph compared to removing it entirely, when possible.
 
 <img src="/resources/images/error-slopes.png" width=700>(Source: Kognity)</img>
 
@@ -186,7 +191,7 @@ Alternatively, for any number of vectors, negate the vector(s) being subtracted
 Vectors can be broken up into two **component vectors** laying on the x- and y-axes via trigonometry such that the resultant of the two components is the original vector. This is especially helpful when adding larger (3+) numbers of vectors.
 $$\vec{F}_x + \vec{F}_y = \vec{F}$$
 
-!!! info "Reminder"
+!!! reminder
     The **component form** of a vector is expressed as $(|\vec{a}_x|, |\vec{a}_y|)$
 
 <img src="/resources/images/vector-simple-adding.png" width=700>(Source: Kognity)</img>
@@ -232,32 +237,86 @@ $$\vec{v} × s = (|\vec{v}|×s)[\theta_{v}]$$
 
 ## 2.1 - Motion
 
+!!! definition
+    - **Uniform motion**: Constant speed.
+    - **Position**: The location of an object relative to an origin (typically the position of the object at time zero).
+    - **Distance**: The scalar of the magnitude of the exact path taken by an object from an initial to a final position.
+    - **Displacement**: The vector of the shortest path from an initial to a final position.
+    - **Acceleration**: The vector of the rate of change of *velocity* over time.
+
 ### Models
 
-A **scientific model** is a simplification of a system based on assumptions used to explain or make predictions for that system.
+A **scientific model** is a simplification of a system based on assumptions that predicts and/or explain observations for that system.
 
 !!! note "Definition"
     - **System**: An object or a connected group of objects.
     - **Point particle assumption**: An assumption that models a system as a blob of matter. It is more reliable if the size and shape of the object(s) do not matter much.
-    - **Uniform motion**: The type of motion in which the speed of an object is constant.
+
+### Velocity
+
+Velocity is the vector of the rate of change of *displacement* over time, and can be represented as $\frac{\Delta d}{\Delta t}$.
+
+The *average* velocity of an object is the velocity over an interval in time, calculated by finding the slope of the **secant** from the start and end position on a position-time graph.
+
+The *instantaneous velocity* of an object is the velocity at a specific moment in time, calculated by finding the slope of the **tangent** at that moment on a position-time graph.
+
+!!! definition
+    - A **secant** is a straight line which intersects two points on a curve.
+    - A **tangent** is a straight line that does not intersect a curve but "touches" it at exactly one point.
 
 ### Displaying motion
 
-Motion can be expressed visually using a **motion diagram** or a **position-time graph**.
-
-// TODO: insert motion diagram here because kognity bad
-
 A **position-time graph** expands on the motion diagram by specifying a precise **position** value on the vertical axis in addition to time on the horizontal axis. The line of best fit indicates the object's speed, as well as if it is accelerating or decelerating.
 
-<img src="/resources/images/position-time-graph.png" width=700>(Source: Kognity)</img>
+$s$ is commonly used in IB to represent displacement and $s_{0}$ represents the initial position (when $t=0$).
 
-When the slope is:
+<img src="/resources/images/position-time-graph.png" width=700>(Source: Kognity)</img>.
 
- - linear, the object is moving at a constant speed.
- - exponential, the object is accelerating.
- - logarithmic, the object is decelerating.
+The slope of the line in a position-time graph represents that object's velocity. If the slope is not linear, the object is not moving uniformly (at a constant speed).
 
-## 2.2 - Forces
+A **velocity-time graph** is similar to a position-time graph but replaces the position on the vertical axis with an object's velocity instead.
+
+<img src="/resources/images/velocity-time-graph.png" width=700>(Source: Kognity)</img>
+
+On a velocity-time graph, the slope represents that object's acceleration. If the slope is not linear, the object is not accelerating uniformly (accelerating at a constant rate).
+
+The area below a velocity-time graph at a given time is equal to the displacement (change in position, $\Delta d$) at that time, since $ms^{-1}×s=m$. When finding the displacement of an object when it is accelerating, breaking up the graph into a rectangle and a triangle then adding their areas will give the displacement.
+
+<img src="/resources/images/velocity-time-displacement.png" width=700>(Source: Kognity)</img>
+
+An **acceleration-time graph** is similar to a velocity-time graph but replaces the velocity on the vertical axis with an object's acceleration instead.
+
+The area below an acceleration-time graph at a given time is equal to the change in velocity ($\Delta v$) at that time.
+
+!!! note
+    If there is any instantaneous jump on a position-, velocity-, or acceleration-time graph (which is impossible in reality but may be used to simplify matters), a dashed line must be used to connect the two sides to ensure that the line remains a function.
+
+### Uniformly accelerated motion
+
+**Uniformly accelerated motion** is the constant acceleration in a **straight line**, or the constant increase in velocity over equal time intervals. The five key $suvat$ variables can be used to represent the various information in uniformly accelerated motion.
+
+### Kinematic equations
+
+<img src="/resources/images/constant-acceleration.png" width=700>(Source: Kognity)</img>
+
+ - $s=$ change in displacement during time interval $t$ (i.e., from $t=0$ to $t$)
+ - $u=$ initial velocity at time $t=0$
+ - $v=$ final velocity at time $t$
+ - $a=$ constant acceleration
+ - $t=$ time elapsed since $t=0$
+
+By the formula of the gradient and the formula for the area underneath an acceleration time graph, the following formulas can be derived and are in the data booklet:
+
+ - $s=ut + \frac{1}{2}at^2$
+ - $v = u + at$
+ - $s = \frac{1}{2}(u+v)t$
+ - $v^2 = u^2 + 2as$
+
+### Projectile motion
+
+**Projectile motion** is uniformly accelerated motion that does not leave the vertical plane (is two-dimensional). Note that the two directions (horizontal and vertical) that the projectile moves in are independent of one another. This means that variables such as average velocity can be calculated by breaking up the motion into the horizontal and vertical axes, then recombined using the Pythagorean theorem such that $v^2 = v_x^2 + v_y^2$.
+
+##2.2 - Forces
 
 ## 2.3 - Work, energy, and power