# Mechanics

calculus-based physics for scientists and engineers

## The Position Function

### The Position Function in One Dimension

The position function is a vector function of time. It is a rule, that for any time (scalar quantity) it will provide a unique vector associated with that time. (If this terminology is unfamiliar, there is a video reviewing functions in the background section.) So, as time varies continuously, the position function provides a continuously varying vector that identifies the location of an object.

### Position Function Terminology

A lot of books assume this is obvious, which it is not. If you have had a physics course before, this is going to be review. However, if not, students have difficulty when all this jargon is delivered and it is assumed everyone knows it.

In the video I show where all this comes from, but for a summary, if you assume a one-dimensional coordinate system of variable, $$x$$:

•   Symbol
• $$x(t)$$
• $$t_i$$
• $$t_f$$
• $$\Delta t$$
• $$x_i$$
• $$x_f$$
• $$\Delta x$$
• $$x_o$$
•   Name
•   position as a function of time
• initial time
• final time
• time difference
• initial position
• final position
• position difference
• initial position, also
•   Expression
•
•
•
• $$t_f-t_i$$
• $$x(t_i)$$
• $$x(t_f)$$
• $$x_f-x_i$$
• $$x(0)$$

Note the different things represented here. There are functions (like, $$x(t)$$), functions evaluated at specific values (like, $$x(t_i)$$), symbols that represent functions evaluated at specific values (like, $$x_i$$), and symbols that represent mathematical combinations of other symbols (like, $$\Delta x$$),

### Distance and Displacement in One Dimension

Now we introduce two new terms: Distance and Displacement. They are different and you must know what those differences are.

First, there must be a well-defined time interval, $$\Delta t$$

Distance is a scalar that is the total length covered during the time interval.

Displacement is a vector that is the position difference, the final position minus the initial position: $$\Delta x = x_f-x_i$$.

### Solution

1. The position function as a function of time of a particle is given by $$x(t) = 10 - 3t$$. Make a tabular representation of the data for each second between $$t_i=0$$ and $$t_i=5$$. Draw a schematic representation of the motion with a coordinate system. Draw out the vectors for each second from your table. Draw a graph of the position as a function of time. What is the distance and the displacement?