Meaning of midpoint clock time

Suppose we have the following angular position vs. clock time data:

This data gives us two intervals.  How long does the first last, and how long does the second?

A typical answer for the first is that it lasts 11 counts, while the second lasts 19 counts.

• However the first interval starts at the count of 1.  A count hasn't elapsed until we have reached 2.  If you think a bit you see that 11 counts don't elapse during this interval, but only 10.
• In any case, note that we naturally subtract 11 from 30 to get the 19 counts in the second interval, so to be consistent we need to subtract the 1 from the 11 to get the number of counts in the first.

Another question:  What is the count at the midpoint of the first interval?

A frequent answer is 5.  However 5 is closer to 1 than to 11, so it isn't halfway.  And 6 is closer to 11 than to 1. 

• The correct halfway count would be the number halfway between 1 and 11, which is found by averaging 1 and 11.
• That is, the halfway count would be (1 + 11) / 2 = 5.5.

Similarly we quickly see that the halfway clock time for the second interval is (11 + 30) / 2 = 20.5.  You can (and should) easily verify that this is equally close to 11 and 30.

Now we ask about the table of average rate of change of angular position vs. midpoint clock time.

We easily see than during the first interval the angular position changes by 180 deg, during an interval that lasts 10 counts, so the desired average rate is 18 deg / count.

Similarly during the second interval the the angular position changes by another 180 deg, during an interval that lasts 19 counts, so the desired average rate is 180 deg / (19 counts)  = 9.5 deg / count.

So our midpoint clock times are 5.5 counts and 20.5 counts, our average angular velocities 18 deg / count and 9.5 deg / count.

Our table is therefore as follows:

midpt clock time (counts)	roc of angular pos wrt clock time (degrees/sec)
5.5	18
20.5	9.5

The logic of this table is simple enough.  The average velocity on an interval typically occurs near (though usually not precisely at) the midpoint clock time.  So this table is a reasonable representation of angular velocity vs. clock time.

We could easily construct a graph of this information, giving us a good approximation of a graph of the actual angular velocity vs. clock time graph.