We will begin our study of dynamics by discussing the kinematics of a particle that moves along a rectilinear or straight-line path. Recall that a particle has a mass but negligible size and shape. The kinematics of a particle is characterized by specifying, at any given instant, the particle's position, velocity, and acceleration.
The straight-line path of a particle will be defined using a single coordinate axis $s$.
The origin $O$ on the path is a fixed point, and from this point the position coordinate $s$ is used to specify the location of the particle at any given instant. The magnitude of $s$ is the distance from $O$ to the particle, usually measured in meters ($m$) or feet ($ft$), and the sense of direction is defined by the algebraic sign on s.
The displacement of the particle is defined as the change in its position.
For example, if the particle moves from one point to another, the displacement is
$$ \Delta s = s' - s $$
In this case $Δs$ is positive since the particle's final position is to the right of its initial position, i.e., $s' > s$. Likewise, if the final position were to the left of its initial position, $Δs$ would be negative.
The displacement of a particle is also a vector quantity, and it should be distinguished from the distance the particle travels. Specifically, the distance traveled is a positive scalar that represents the total length of path over which the particle travels.
If the particle moves through a displacement $Δ$s during the time interval $Δt$, the average velocity of the particle during this time interval is
$$ v_{avg} = \frac{\Delta s}{\Delta t} $$
If we take smaller and smaller values of $Δt$, the magnitude of $Δs$ becomes smaller and smaller. Consequently, the instantaneous velocity is a vector defined as: $v = \lim_{\Delta t \to 0} (\Delta s / \Delta t)$, or