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 velocity v 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 t 0 is also known as the instantaneous velocity of the object at time t = t 0 , 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 (t 0, t 1) is given by:
In other words, the average velocity is the total displacement divided by the total time. Notice that if a car leaves its garage in the morning, drives all around town throughout the day, and ends up right back in the same garage at night, its displacement is 0, meaning its average velocity for the whole day is also 0.
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:
This is, in fact, the formula for the velocity function in terms of the position function! (In the language of calculus, this is also known as the formula for the derivative of x with respect to t .) Unfortunately, it is not feasible, in general, to compute this limit for every single value of t. However, the position functions we will be dealing with in this SparkNote (and those you will likely have to deal with in class) have exceptionally simple forms, and hence it is possible for us to write down their corresponding velocity functions in terms of a single rule valid for all time. In order to do this, we will borrow some results from elementary calculus. These results will also prove useful in our discussion of acceleration.