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that time “goes backward” in the sense that adults can shrink into infants
and retreat into the womb. It just means that we have to pick a reference
point and call it t=0, and then times before that are represented by negative
values of t.
Although a point in time can be thought of as a clock reading, it is
usually a good idea to avoid doing computations with expressions such as
“2:35” that are combinations of hours and minutes. Times can instead be
expressed entirely in terms of a single unit, such as hours. Fractions of an
hour can be represented by decimals rather than minutes, and similarly if a
problem is being worked in terms of minutes, decimals can be used instead
of seconds.
Self-Check
Of the following phrases, which refer to points in time, which refer to time
intervals, and which refer to time in the abstract rather than as a measurable
number.
(a) “The time has come.”
(b) “Time waits for no man.”
(c) “The whole time, he had spit on his chin.”
Position as opposed to change in position
As with time, a distinction should be made between a point in space,
symbolized as a coordinate x, and a change in position, symbolized as
.
x.
As with t, x can be negative. If a train is moving down the tracks, not
only do you have the freedom to choose any point along the tracks and call
it x=0, but it’s also up to you to decide which side of the x=0 point is
positive x and which side is negative x.
Since we’ve defined the delta notation to mean “after minus before,” it
is possible that
.
x will be negative, unlike
.
t which is guaranteed to be
positive. Suppose we are describing the motion of a train on tracks linking
Tucson and Chicago. As shown in the figure, it is entirely up to you to
decide which way is positive.
Enid
x=0
Chicago
Tucson
x>0
x<0
.
x>0
Joplin
Enid
x=0
Chicago
Tucson
x<0
x>0
.
x<0
Joplin
Two equally valid ways of describing the motion of a train from Tucson to
Chicago. In the first example, the train has a positive
.
x as it goes from Enid to
Joplin. In the second example, the same train going forward in the same
direction has a negative
.
x.
(a) a point in time; (b) time in the abstract sense; (c) a time interval
Chapter 2Velocity and Relative Motion