Position, Velocity, and Acceleration
Position
Definition
The function representing an object’s position is referred to as the position function, or simply position.
Description
In physics, since we consider the change of position over time (referred to as motion), position is a function of time, as shown by $x = x(t)$. When assuming a one-dimensional space, it is commonly denoted as $x$. In the case of two-dimensional or three-dimensional spaces, it is usually denoted in boldface r as $\mathbf{r}$, and called a position vector. Mathematically, it is expressed as follows:
$$ \mathbf{r} : [0, \infty) \to \mathbb{R}^{n},\quad n=1,2,3 $$
The following are commonly used notations for position vectors in three-dimensional space. When not explicitly expressing the position function, the variable $(t)$ is often omitted.
$$ \begin{align*} \mathbf{r} &= (x, y, z) \\ &= x\hat{\mathbf{x}} + y\hat{\mathbf{y}} + z\hat{\mathbf{z}} \\ &= x\hat{\mathbf{e}_{1}} + y\hat{\mathbf{e}_{2}} + z\hat{\mathbf{e}_{3}} \\ &= x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \end{align*} $$
On a blackboard, an arrow is usually used to denote as $\vec{r}$, but textbooks often use boldface to denote as $\mathbf{r}$. When drawing a position-time graph, it is common to set the horizontal axis to time $t$ and the vertical axis to position $x$.
Displacement
The change in position from time $t_{0}$ to $t_{1}$ is referred to as displacement. It’s a term not often used.
$$ \Delta \mathbf{r} = \mathbf{r}(t_{1}) - \mathbf{r}(t_{0}) $$
Since we only consider the difference between times $t_{0}$ and $t_{1}$, having a displacement of $0$ does not mean there was no movement during that time.
Velocity
Definition
The derivative of the position function is referred to as velocity, denoted as $\mathbf{v}$.
$$ v(t) = \dfrac{d x(t)}{d t} $$
$$ \mathbf{v}(t) = \dfrac{d \mathbf{r}(t)}{d t} $$
Description
If velocity is a constant function, it is said that the object is in uniform motion.
Integration is the inverse operation of differentiation, therefore the following holds true:
$$ x(t_{1}) = x(t_{0}) + \int_{t_{0}}^{t_{1}} v(\tau) d\tau $$
Average Velocity
The average velocity is obtained by dividing the displacement by the motion time $\Delta t = t_{1} - t_{0}$.
$$ \overline{v} = \dfrac{\Delta x}{\Delta t} $$
$$ \overline{\mathbf{v}} = \dfrac{\Delta \mathbf{r}}{\Delta t} $$
Speed
The magnitude $\left| \mathbf{v} \right|$ of velocity is referred to as speed. Personally, I believe the term “speed” should not be used. It’s not particularly an important concept and just the magnitude of velocity is sufficient. It only serves to confuse those learning physics for the first time, which is not beneficial.
度 (도) means degree, indicating a scalar, and 力 (력) means force, indicating a vector, so there is an opinion that the translations of velocity and speed are incorrect.
Acceleration
Definition
The derivative of the velocity is referred to as acceleration, denoted as $\mathbf{a}$.
$$ a(t) = \dfrac{d v(t)}{d t} $$
$$ \mathbf{a}(t) = \dfrac{d \mathbf{v}(t)}{d t} $$
Description
Acceleration is the second derivative of the position function. Since integration is the inverse operation of differentiation, the following holds true:
$$ v(t_{1}) = v(t_{0}) + \int_{t_{0}}^{t_{1}} a(\tau) d\tau $$
If acceleration is a constant function, it is said that the object is in uniform acceleration motion. Including advanced physics, most physical situations we learn assume uniform acceleration motion.