Position and Velocity

How can motion be described using changes in position and time?

When an object moves, its position changes during (or over) a period of time.  You can think of position as a certain mark along a number line that you might be standing at.

Figure 1

1st part (1s to 5s)

In the diagram above, a girl starts at an initial position of 8m.  Some time later she is at a final position of 34m.  Her change of position is 26m and this can be calculated as shown below.

Example 3.1.1

final position

initial position

change in position

The time at the starting position was zero hours, 12 minutes, and 2 seconds.  The time at the final position was zero hours, 12 minutes, and 48 seconds.  So the change in time (∆t) is 46 seconds.  The change in time is the period of time that elapses during her change in position. 

A period of time is different from a instant in time.  An instant in time has a certain reading on the clock, such as 00:12:34.  The instant starts at 00:12:34 and ends at 00:12:34.  This instant is a period of time of zero seconds.  You could say an instant is infinitely ‘short’.

How can we use this information to calculate the velocity of the girl?  The velocity of the girl is how much her position changes during a certain period of time.  This can be expressed in a ratio of ∆x over ∆t.

Example 3.1.2

This velocity can also be expressed on a graph with position on the y-axis and time on the x-axis, as seen below on the position-time graph.  Click here (EWS)to see a video demo of a cylinder rolling at constant velocity.

The graph above represents the motion of the girl from Figure 1.  As in Figure 1, the girl’s starting position is 2 meters and her final position is 48 meters.  The starting time on the clock is 2 seconds and the final time on the clock is 48 seconds.

These points or positions at certain times can be represented in x-y coordinates, which you have seen in your math courses, such as [2s, 8m] for the first point on the line and [48s, 34m] for the last point on the line, as shown on the graph below.  Remember these coordinates have units. The x-coordinate is the time and the y-coordinate is the position.

Let’s calculate the slope of this line.  You know that slope is m = ∆y/∆x and this can be expressed as

Figure 2

To calculate the slope of the line, we take Equation 4 and substitute in the relevant positions and times.

Figure 3

On this graph, however, the x-coordinate is really the time (t) and the y-coordinate is really the position (x).  So on this graph, the slope is really m = ∆x/∆t.  This can be expressed as

[2,8]

[48,34]

Equation 1

Equation 2: This is the velocity       

                equation!

Equation 3

Equation 4

Example 3.1.3

The slope is equal to the velocity !

The slope we calculated is equal to the velocity that we calculated in example 3.1.2, which means the slope on a position-time graph IS the velocity of the object!

Let’s see other examples of motion expressed on a graph.  On the graph below, a girl starts at a position of 5 meters at a time of 1 second.  She moves to the 25 meter position, stays there for sometime, moves to the 40 meter position, then moves back to the 15 meter position.  All this happens as time passes.  Let’s calculate the girl’s velocity during different portions of her trip.

Example 3.1.4

2nd part (5s to 9s)

3rd part (9s to 15s)

4th part (15s to 19s)

According to our results, the girl was moving relatively quickly during the first part of the trip,  she was not moving during the second part of the trip and she was moving slower during the third part of the trip.  How do you describe the way she was moving during the last part of the trip?  The answer is she was moving backwards and relatively quickly backwards!  It is important to remember that the negative sign on the velocity means an object is moving backwards.  If you drew a diagram of the girl moving, similar to Figure 1, what would it look like?  Click here to see.

To find the overall velocity or average velocity (to be discussed more in the next section) over the entire trip, we would use the same formula.

Figure 4

Example 3.1.5

The overall velocity (or average velocity of the whole trip) is small because the girl spent part of the trip not moving and part of the trip going backwards.

Compare these graphs and calculations to those you made in the Motion Graphs Lab.

Copyright  © 2009-2012, by Marcus Milling

edited by Julia Pian