# Time

#### Prerequisites

Describing motion is all about "where and when". The first half of talking about motion is specifying "where" — choosing a reference point and setting up a coordinate system.

The other half of talking about motion is specifying "when". For this we need to quantify time — create a mathematical model of it.

## Modeling time

We experience lots of regularly repeating events in our daily lives — 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 ECGs, clocks, and calendars. The regularity of repetition helps us develop our sense of time.

But our everyday descriptions are not particularly convenient for doing science because of the messy units — seconds, hours, days, weeks, months, years, etc. 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.

## Creating a time coordinate

To create a mathematical model of time for use in science, we map time onto the real number line. Just as we did with creating a spatial coordinate system, we have to make a number of choices.

• Pick an origin. (Choose a time to call "0".)
• Pick a positive direction. (We usually 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 do to the inconvenient conversion factors.)

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

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 we will want the other. If we're not careful this can lead to confusion.

For this reason, you need to pay careful attention to the difference between the value of time ($t$) and a time interval (change in time $\Delta t$). 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. Keeping the $\Delta$ whenever you are talking about a time interval is a very good idea. (See Values, change, and rates of change.)

Joe Redish 7/25/11

Article 310