Kinematics
Today, we are going to talk about how physicists describe motion. The ingredients to the description we will talk about are
Position is a concept used by everyone. Reading a map (see below) is probably the most familiar example. Even so, we still need to give a careful physics definition.
The ingredients to a position measurement are
- the definition of an origin. In the above map, Bloomington is the origin.
- the definition of coordinates axes.
These are the perpendicular lines on the map, pointing in the north/south direction (which I've called the y axis) and in the east/west direction (which I've called the x axis).
To specify the position of Indianapolis in our coordinate system requires two measurements. One measurement is how far Indianapolis is to the east of Bloomington (35 km) and the other is how far Indianapolis is to the north of Bloomington (67 km). That set of numbers, (x,y) = (35 km, 67 km) is called the position of Indianapolis. Using the Pythagorus theorem, we can also specify the position by the straight line distance between Bloomington and Indianapolis and the direction.
Motion can be represented on a map (or a position graph) by a trajectory, which is nothing more than the path followed by an object. Points along the trajectory can be measured positions, with the smooth line connecting the measurements. In the picture above, Bob takes a circuitous path from Evansville to Fort Wayne.
There is another measurement, beyond a set of position measurements, that we need to quantitatively describe motion . That quantity is time: when were the position measurements made? I've taken away the Indiana state landmarks and left only a position graph showing Bob's trajectory. Now, I've shown as dots, the points on the trajectory where position measurements were made and have also shown the measured time for each point in hours. We can now start making qualitative statements about the motion: Bob was moving more quickly between hours 5 and 6 than he was between hours 1 and 2.
To be more quantitative about motion, it is very convenient to first simplify things. Let's consider Bob moving along a straight line, with Bloomington at the origin. For the sake of argument, let's say he moves only east or west. We'll call positive x to the east and negative x to the west. Below is a table of position measurements:
| x (km) | Time (hours)
|
| +20 | 0
|
| +10 | 1
|
| 0 | 2
|
| 0 | 3
|
| -10 | 4
|
| +20 | 5
|
| +40 | 6
|
Between the 2nd and 3rd hour Bob wasn't moving. At other times he was in motion. To analyze his motion, it is very convenient to draw a position versus time graph. First, we plot the measured points and then we draw a smooth curve which represents his positions for the in-between times.
The average velocity is the ratio of the change in position to the time interval over which the position changed. We write this mathematically as
There are several things to note
- When the velocity is negative, Bob is heading to the west. On the graph, this is represented by lines heading downward.
- When the velocity is positive, Bob is heading to the east. On the graph, this is represented by lines heading upward.
- The steeper the line is on the graph, the faster Bob is moving.
This is just the beginning at understanding the utility of graphical representations of motion!