- The leftmost non-zero digit is the **most significant digit**.
- If there is no decimal point, the rightmost non-zero digit is the **least significant digit**.
- Otherwise, the right-most digit (including zeroes) is the least significant digit.
- All digits between the most and least significant digits are significant.
- Pure (discrete) numbers are unitless and have infinite significant figures.
!!! example
In $123000$, there are 3 significant digits.<br>
In $0.1230$, there are 4 significant digits.
- When adding or subtracting significant figures, the answer has the **same number of decimals** as the number with the lowest number of decimal points.
- When multiplying or dividing significant figures, the answer has the **same number of significant figures** as the number with the lowest number of significant figures.
- Values of a calculated result can be **no more precise** than the least precise value used.
When rounding an answer with significant figures, if the **least significant figure** is $5$, round up only if the **second-least** significant figure is **odd**.
Scientific notation is written in the form of $m×10^{n}$, where $1 \leq m <10,n \in \mathbb{Z}$.Alldigitsbeforethemultiplicationsigninscientificnotationaresignificant.<br>
!!! example
The speed of light is 300 000 000 ms<sup>-1</sup>, or $3×10^8$ ms<sup>-1</sup>.
### Orders of magnitude
The order of magnitude of a number can be found by converting it to scientific notation and taking its power of 10.
| Caused by imperfect measurements and is present in every measurement. | Caused by a flaw in experiment design or in the procedure. |
| Can be reduced (but not avoided) by repeated trials or measurements. | Cannot be reduced by repeated measurements, but can be avoided completely. |
| Error in precision. | Error in accuracy. |
!!! example
- The failure to account for fluid evaporating at high temperatures is a systematic error, as it cannot be minimised by repeated measurements.
- The addition of slightly more solute due to uncertainty in instrument data is a random error, as it can be reduced by averaging the result of multiple trials.
Uncertainties are stated in the form of $a±\Delta a$. A value is only as precise as its absolute uncertainty. Absolute uncertainty of a **measurement** is usually represented to only 1 significant digit.
- a digital instrument is used, the last reported digit is uncertain by 1 at its order of magnitude.
!!! example
- A ruler has millimetre markings. A pencil placed alongside the ruler has its tip just past 14 mm but before 15 mm. The pencil is 14.5 mm ± 0.5 mm long.
- A digital scale reads 0.66 kg for the mass of a human body. The human body has a mass of 0.66 kg ± 0.01 kg.
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 $(x ± \Delta x, y ± \Delta y)$
Vectors are drawn as arrows whose length represents their scale/magnitude and their orientation refer to their direction. A variable representing a vector is written with a right-pointing arrow above it.
When subtracting exactly one vector from another, repeat the steps above, but instead place the second vector at the **tail** of the first, then draw the resultant vector from the head of the second vector to the head of the first vector. Note that this only applies when subtracting exactly one vector from another.
!!! example
In the diagram above, $\vec{b}=\vec{a+b}-\vec{a}$.
Alternatively, for any number of vectors, negate the vector(s) being subtracted by **giving it an opposite direction** and then add the negative vectors.
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.
The product of a vector multiplied by a scalar is a vector with a magnitude of the vector multiplied by the scalar with the same direction as the original vector.
- **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.
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.
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.
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).
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.
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.
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** 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.
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:
**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 vector **components**, then finding the resultant vector.
A force is a "push" or "pull" exerted by one object acting on another object and is always represented in the form of a vector. The SI unit for force is the newton ($\text{N}$), equivalent to $\text{kg} \cdot \text{m} \cdot \text{s}^{-2}$.
Forces do not cause motion but instead cause *changes* in motion; therefore a constant force on an object results in uniformly acceleration motion**.
A force must have:
- A magnitude and direction
- An object on which it acts and a location on that object where it acts
- An object providing the force
- An object acting and an object acted upon that are *not* the same
A force must be classified as one of the following types of forces:
Contact (the two objects are touching) forces:
- **Tension** ($\vec{F_T}$) is the pulling force from one object on another. Note that tension goes both ways: whichever object is pulling is also being pulled in the opposite direction.
- **Normal force** ($\vec{F_N}$) is the "resistance" or "pushback" force from one object on another pushing on it. It is always perpendicular to the surfaces of the objects.
- **Kinetic and static friction** ($\vec{F_{fK}}$ and $\vec{F_{fS}}$, respectively) is the force resisting the motion of two surfaces sliding against each other. Kinetic friction opposes an object in motion while static friction opposes an object from moving in the first place. The vector of friction is parallel to the surface and opposite the direction of motion.
- **Drag** ($\vec{F_D}$) is the resistive force caused by fluids in the path of an object. Its vector is always pointing in the opposite direction of the object's *instantaneous* velocity.
- **Thrust** ($\vec{F_T}$) is the propulsive force caused by mass (usually fluids) being ejected from an object. The force vector always points opposite the direction of the ejected mass.
Non-contact forces:
- **Weight** ($\vec{W}$) is the attraction between a mass and a planet. Its vector always points toward the planet.
- **Electromagnetic forces** ($\vec{F_M}$ and $\vec{F_E}$) are not covered in this course.
