1.1 Speed and Velocity
Motion is defined as: the change of position with time. The standard units used to describe motion are the meter (m) for measuring change in position, and the second (s) for measuring time. Two quantities are often used to describe the rate of change of position: velocity and speed. Velocity is a vector quantity, while speed is a scalar. What does this mean? Measures of velocity will have a magnitude, or numeric value, and a direction. Speed, on the other hand, has only a numeric value.
This difference between speed and velocity leads to another. Speed is the quotient of total distance traveled and time, while velocity is displacement (total change in position) divided by the time elapsed. For example, a man walks 10.0 meters east, and then 70.0 meters west in 90.0 seconds. His average speed is (10.0 m + 70.0 m) / 90.0 s, or 0.889 m/s. His average velocity, on the other hand, was only 0.667 m/s. Since the man walked 10.0 m east, then turned around, walked 10.0m back to his starting point, and walked an additional 60.0 m west, the total displacement was only 60.0 m. Thus, his average velocity was (70.0 m - 10.0 m) / 90.0 s: 0.667 m/s west [or (10.0 m - 70.0 m) / 90.0 s = -0.667 m/s east].
Velocity, itself, comes in two flavors: average and instantaneous. Average velocity is the quotient of displacement and time elapsed, where the elapsed time is greater than 0. Translated into the language of mathematics, the definition of average velocity is:
where v represents average velocity, s represents position, and t represents time. The Greek letter D (delta) means "change in."
Instantaneous velocity, on the other hand, is defined as the limit of average velocity as D t approaches zero:
The derivative nature of velocity (and acceleration) will become more apparent in the discussion of acceleration, next.