We have already discussed examples of position functions in the previous section. We now turn our attention to velocity and acceleration functions in order to understand the role that these quantities play in describing the motion of objects. We will find that position, velocity, and acceleration are all tightly interconnected notions.
In one dimension, velocity is almost exactly the same as what we normally call speed. The speed of an object (relative to some fixed reference frame) is a measure of "how fast" the object is going--and coincides precisely with the idea of speed that we normally use in reference to a moving vehicle. Velocity in one-dimension takes into account one additional piece of information that speed, however, does not: the direction of the moving object. Once a coordinate axis has been chosen for a particular problem, the velocityv of an object moving at a speed s will either be v = s, if the object is moving in the positive direction, or v = - s, if the object is moving in the opposite (negative) direction.
More explicitly, the velocity of an object is its change in position per unit time, and is hence usually given in units such as m/s (meters per second) or km/hr (kilometers per hour). The velocity function, v(t), of an object will give the object's velocity at each instant in time--just as the speedometer of a car allows the driver to see how fast he is going. The value of the function v at a particular time t0 is also known as the instantaneous velocity of the object at time t = t0, although the word "instantaneous" here is a bit redundant and is usually used only to emphasize the distinction between the velocity of an object at a particular instant and its "average velocity" over a longer time interval. (Those familiar with elementary calculus will recognize the velocity function as the time derivative of the position function.)
Now that we have a better grasp of what velocity is, we can more precisely define its relationship to position.
We begin by writing down the formula for average velocity. The average velocity of an object with position function x(t) over the time interval (t0, t1) is given by:
As the time intervals get smaller and smaller in the equation for average velocity, we approach the instantaneous velocity of an object. The formula we arrive at for the velocity of an object with position function x(t) at a particular instant of time t is thus: