Velocity
In order for you or me to calculate the speed of an object we must know how far it goes and how long it takes to get there. Astute observers should then ask a following question …
What do you mean by “how far”? Didn’t we learn in the previous section that there are two quantities used to answer the question “how far”?
My, but you are wise. Yes indeed, there are two ways to answer that question. When you ask “how far” do you mean distance or displacement? There’s a difference between the two quantities and thus a difference between the two answers.
Speed is the rate of change of distance with time. Velocity is the rate of change of displacement with time.
Which means that for the calculus people …
Speed is the first derivative of distance with respect to time. Velocity is the first derivative of displacement with respect to time.
Velocity and speed mean pretty much the same thing to the average English speaking person, but physics is more precise in its language than is everyday speech.
The situation is not entirely hopeless, however. All the types of speed discussed above also have their counterparts in velocity. Just replace the symbol for distance with the symbol for displacement et voila, you’ve got velocity.
Equations for Speed
Average: 
Instantaneous: 
Equations for Velocity
Average: 
Instantaneous: 
Speed and velocity are related in much the same way that distance and displacement are related. Displacement is measured along the shortest path between two points and thus its magnitude is always less than or equal to the distance. The magnitude of the displacement approaches the distance as distance approaches zero. That is, distance and displacement are effectively the same (have the same magnitude) when the interval examined is “small”. Since speed is based on distance and velocity is based on displacement, these two quantities are effectively the same (have the same magnitude) when the time interval examined is “small” or, in the language of calculus the magnitude of an object’s average velocity approaches its average speed as the time interval approaches zero. Likes that;
Thus, the instantaneous speed of an object is the magnitude of its instantaneous velocity.
v = |v|
Units
Speed and velocity are both measured using the same units. Given that the SI unit of both distance and displacement is the meter and that the SI unit of time is the second, it should be intuitively obvious that the unit of both speed and velocity would be a ratio of two units. The SI unit of speed and velocity is the meter per second [m/s].
This unit is only rarely used outside scientific and academic circles. Most people on this planet measure speeds in kilometer per hour (km/h or sometimes kph). The United States is an exception in that we use the comparatively archaic mile per hour (mi/h or mph). Let’s determine the conversion factors so that we can relate speeds measured in m/s with the more familiar, everyday units.
The decimal values are accurate to four significant digits, but the fractional values should only be considered rules of thumb (1 mph is really more like 4/10 m/s).
The ratio of any unit of distance to any unit of time is a unit of speed.