To the ancient Greeks motion was either natural or violent. The four elements sought their natural locations. Earth (including metals) fell down because it had a property called gravity. Fire (including smoke) went up because it had a property called levity. Water was between Earth and air. Heavenly objects, made of aether, moved in circles.

Arrows or other objects that were thrown were said to move because they were given violent motion. What was violent motion? The bow transferred force to the arrow; the thrower transferred force to the rock. Once in air (or water) the medium pushed the object along. When the force ran out the medium now opposed the motion and the object fell to Earth.

In the sixth century, the commentator Philoponus doubted Aristotle’s view of the role of the medium in motion. Avempace, whose Arabic name was Ibn Bajja, was a Spanish Arab who died in 1138. He also discussed the role of the medium. While Aristotle claimed that motion in a vacuum would be impossible Avempace stated that motion in a vacuum would continue forever because nothing opposed it.

It wasn’t until 1330 that the possibility that motion could vary was suggested. In that period philosophers at Merton College, part of Oxford University, developed their ideas of instantaneous speed and acceleration. Scholars at the University of Paris contributed greatly to these definitions that made the modern measurements of motion possible.

Just as you can add direction to a change in position and end up with displacement, you can also specify the direction of motion. The combination of speed with direction is called velocity. Velocity is the displacement divided by the time required to make the change, or the time rate of change of displacement. Velocity is a vector quantity, like displacement.

You might walk at 4 mph north, or a balloon might move at 5 feet per second up. If you assign the variable x to represent north/south, γ to east/west, and z to up/down position, then both the change in position and speed would be positive for movement to the north, east, or up. Your walking velocity would be 4x mph. The balloon’s would be 5z ft/s.

Average speed is almost always more useful than average velocity. For example, in a NASCAR race the starting and finish lines are in the same position. So, no matter how fast the cars go, their average velocity is zero because the beginning and ending positions are the same. Instantaneous velocity, however, is more useful than instantaneous speed, as will be shown shortly.