Kinematics

Kinematics is the study of a body's motion without concern to the forces that act upon it.

Displacement

A body's displacement is its change in position from some initial point of reference. Displacement is defined the difference between that body's current position $x$ and its initial position $x_0$, such that:

$$s=x-x_0$$

Where $s$ is the displacement of the body, which may also be expressed as $\Delta x$. Additionally, in constant acceleration, a body's displacement may also be expressed as:

$$\begin{array}{cc}s& =v_{0}t+\frac{1}{2}a{t}^2\\ & =\frac{1}{2}(v_0+v)t\end{array}$$

Where $a$ is the acceleration, $v$ is the velocity, $v_0$ is the initial velocity, and $t$ is the time.

Velocity

A body's velocity is the rate at which it is displaced over a given interval of time. Therefore, the velocity $v$ is proportional to the displacement $\Delta x$ and change in time $\Delta t$, such that:

$$v=\frac{\Delta x}{\Delta t}$$

The velocity at a given moment for a body experiencing some acceleration $a$ over a period of $t$ starting at some velocity $v_0$ is given by:

$$v=v_0+at$$

Additionally, when displacement $s$ is known, the velocity is given by:

$$v^2-v_0^2=2as$$

Acceleration

A body's acceleration is the rate at which an object's velocity changes over a given interval of time. Thus, the acceleration $a$ is given by the change in velocity $\Delta v$ over an interval $\Delta t$, such that:

$$a=\frac{\Delta v}{\Delta t}$$