Class content I > The Main Question: Motion > Kinematics
3.1.3
Prerequisites
Modeling time
The other half of talking about motion (after figuring out "where") is specifying "when". For this we need to quantify time  create a mathematical model of it.
There are lots of regularly occurring events in our experience  our heartbeat, day and night, phases of the moon, seasons, etc.  and they give us a personal sense of time and time passing. We've quantified that by creating clocks, but our everyday description turns out to be inconvenient for doing science because of the messy units  years, months, weeks, days, hours, seconds, etc. In order to be able to do math with time, we want to map physical time (the time in the real world) into a mathematical structure that carries with it all the nice properties of arithmetic that match our sense of time: ordering, adding, and subtracting.
To create a mathematical model of time for use in science, we map time onto the real number line. Just as with space, we have to make a number of choices.
 Pick an origin (choose a time to call "0")
 Pick a positive direction (usually we take the future as positive and the past as negative)
 Pick a unit (this can be any one of our standards  second, minute, hour, year  but we shouldn't mix them)
While this idea of mapping time onto the real number line seems quite reasonable, it turns out to be tricky to use. Each point on the number line of time corresponds to a particular "fixed instant" of time. Thinking about such a fixed instant is "stopping time". It's like we are watching a movie and a particular time corresponds to specifying a particular frame of the movie. Typically we tend to think about what's happening as a continuous and related set of events. "Stopping time" and thinking about what is happening at a particular instant (while time continues to run in our heads!) can be more challenging than you expect.
Clock time and time intervals
A second issue about time that turns out to be easy to forget is that we will be interested both in when something happens (the clock time) and how long it takes (the time interval). In some physical situations we will want one, in other situations the other. If we're not careful this can lead to confusion. For this reason, we will pay attention to the difference between the value of time (t) and the change in time (Δt). Unfortunately, some textbooks are sloppy about this in writing equations. Until you are very comfortable with equations, and are blending the symbolic and the physical meaning easily and quickly, be careful to make the distinction.


Followons
Joe Redish 7/25/11
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