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Update 'docs/sph3u7.md'

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magicalsoup 2020-09-24 19:15:30 -04:00
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@ -36,6 +36,9 @@ Every SI unit can be expanded with metric prefixes.
| peta- | P | $10^{15}$ | f | femto- | | peta- | P | $10^{15}$ | f | femto- |
| exa- | E | $10^{18}$ | a | atto- | | exa- | E | $10^{18}$ | a | atto- |
!!! note
For easier memorization, notice that most of these prefixes are in multiples of $3$.
### Significant figures ### Significant figures
- The leftmost non-zero digit is the **most significant digit**. - The leftmost non-zero digit is the **most significant digit**.
@ -112,7 +115,7 @@ Uncertainties are stated in the form of [value] ± [uncertainty]. A value is onl
To determine a measurement's absolute uncertainty, if: To determine a measurement's absolute uncertainty, if:
- the instrument states its uncertainty, use that. - the instrument states its uncertainty, use that.
- an analog instrument is used, the last digit is estimated and appended to the end of the reported value. The estimated digit is uncertain by 5 at its order of magnitude. - an analog instrument is used, the uncertainty is half of the smallest scale. (eg if a ruler can measure to $0.1cm$, the uncertainty would $0.005 cm$)
- a digital instrument is used, the last reported digit is uncertain by 1 at its order of magnitude. - a digital instrument is used, the last reported digit is uncertain by 1 at its order of magnitude.
!!! example !!! example
@ -142,9 +145,9 @@ Error bars represent the uncertainty of the data, typically representing that da
<img src="/resources/images/error-slopes.png" width=700>(Source: Kognity)</img> <img src="/resources/images/error-slopes.png" width=700>(Source: Kognity)</img>
The uncertainty of the **slope** of the line of best fit is the difference between the maximum and minimum slopes. The uncertainty of the **slope** of the line of best fit is the difference between the maximum and minimum slopes.
$$m_{best fit} ± m_{max}-m_{min}$$ $$m_{\text{best fit}} ± \dfrac{m_{\max}-m_{\min}}{2}$$
The uncertainty of the **intercepts** is the difference between the intercepts of the maximum and minimum lines. The uncertainty of the **intercepts** is the difference between the intercepts of the maximum and minimum lines.
$$intercept_{best fit} ± intercept_{max} - intercept_{min}$$ $$\text{intercept}_{\text{best fit}} ± \dfrac{\text{intercept}_{\max} - intercept_{\min}}{2}$$
## 1.3 - Vectors and scalars ## 1.3 - Vectors and scalars
@ -166,6 +169,9 @@ $$\vec{a} = (1, 1)$$
- The **magnitude** of a vector can be expressed as the absolute value of a vector. - The **magnitude** of a vector can be expressed as the absolute value of a vector.
$$|\vec{a}| = 1 \text{ m}$$ $$|\vec{a}| = 1 \text{ m}$$
!!! info "Reminder"
Remember that the **minimum magnitude** for any vector must be $`\ge 0`$
### Adding/subtracting vectors diagrammatically ### Adding/subtracting vectors diagrammatically
1. Draw the first vector. 1. Draw the first vector.
@ -179,6 +185,9 @@ When subtracting a vector, **negate** the vector being subtracted by giving it a
<img src="/resources/images/vector-subtract-direction.png" width=700>(Source: Kognity)</img> <img src="/resources/images/vector-subtract-direction.png" width=700>(Source: Kognity)</img>
!!! note
Notice that when we are subtracting vectors ($\vec{a} - \vec{b}$), we have **tail** to **tail** and the difference vector has a direction from ($\vec{b} to \vec{a}$)
### Adding/subtracting vectors algebraically ### Adding/subtracting vectors algebraically
Vectors can be broken up into two vectors (**"components"**) 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. Vectors can be broken up into two vectors (**"components"**) 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.