Definition of Speed
The speed of a body is defined by the change in its position within an elapsed time interval. If it is at position $s_1$ at a time $t_1$ and at position $s_2$ at a different time $t_2$, then its velocity $v$ (from English velocity) can be calculated as follows:
$$v = \frac{s_2 - s_1}{t_2 - t_1} = \frac{\Delta s}{\Delta t}$$
Example: The driving distance between the two German cities Cologne and Frankfurt was determined to $\Delta s = 220\,\mathrm{km}$. If the average speed of the car is $v=110\,\mathrm{km/h}$, the required time is given as $$\Delta t = \frac{\Delta s}{v} = \frac{220\,\mathrm{km}}{110\,\mathrm{km/h}} = 2\,\mathrm{h}$$
Here, a unit conversion into SI units was not required because the given values and the expected results have the same units km/h.
Experiment
The velocity can be measured using an air-cushion glider that moves frictionlessly along a rail after being given a push. Using two light barriers, the elapsed time is measured for different distances. The quotient of the two quantities, defined as velocity, turns out to be constant. Note that the equation mentioned above only applies if the body moves at a constant speed during the time interval under consideration, i.e., if it is moving in a straight line and at a uniform speed.
Momentary Speed
If its speed changes at any time $t$, then the equation describes the average speed, which is usually denoted by $\bar{v}$. In this case, it is referred to as accelerated motion. To determine the instantaneous velocity of an accelerating body, the time interval must be chosen as small as desired. Mathematically, this corresponds to the derivative with respect to time.
Momentary Speed
The momentary speed of an object with a distance $s(t)$ as a function of the time $t$ can be calculated according to:
$$v = \lim_{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t} = \frac{\mathrm{d}s}{\mathrm{d}t} = \dot{s}$$
Unit Conversion
To calculate velocity analytically, the distance must be given as a function of time. The unit of speed is the quotient of distance and time and is a combination of SI units, meters per second (m/s). Another very popular unit is kilometers per hour (km/h), which is used, for example, in road traffic and for measuring wind speed. The relationship between these two units is as follows:
$$1\,\mathrm{\frac{km}{h}} = 1,000\,\mathrm{\frac{m}{h}} = \frac{1,000\,\mathrm{m}}{3,600\,\mathrm{s}} = \frac{1}{3.6}\frac{\mathrm{m}}{\mathrm{s}}$$
To obtain the unit km/h, the value in m/s must be multiplied by a factor of 3.6, i.e., the following applies:
$$v\left[\frac{\mathrm{km}}{\mathrm{h}} \right] = 3.6\cdot v\left[\frac{\mathrm{m}}{\mathrm{s}} \right]$$
Definition of Velocity
In general, a body does not move along an axis, but rather on curvilinear paths in three-dimensional space. Instead of the position $x$, the position vector $\vec{r} = (x, y, z)$ will be used, which includes all three coordinates. In this case, the velocity must also be treated as a vector quantity.
Velocity
The velocity of an object with a position described by the vector $\vec{r}$ is given as:
$$\vec{v} = \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} =\begin{pmatrix}\dfrac{\mathrm{d}x}{\mathrm{d}t}\\ \dfrac{\mathrm{d}y}{\mathrm{d}t}\\ \dfrac{\mathrm{d}z}{\mathrm{d}t}\end{pmatrix}$$
Each vector component of the position vector $\vec{r}$ must be differentiated with respect to time $t$ independently of the other two components. The result is the velocity vector $\vec{v}$.