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Note that in addition to x and
.
x, there is a third quantity we could
define, which would be like an odometer reading, or actual distance
traveled. If you drive 10 miles, make a U-turn, and drive back 10 miles,
then your
.
x is zero, but your car’s odometer reading has increased by 20
miles. However important the odometer reading is to car owners and used
car dealers, it is not very important in physics, and there is not even a
standard name or notation for it. The change in position,
.
x, is more useful
because it is so much easier to calculate: to compute
.
x, we only need to
know the beginning and ending positions of the object, not all the informa-
tion about how it got from one position to the other.
Self-Check
A ball hits the floor, bounces to a height of one meter, falls, and hits the floor
again. Is the
.
x between the two impacts equal to zero, one, or two meters.
Frames of reference
The example above shows that there are two arbitrary choices you have
to make in order to define a position variable, x. You have to decide where
to put x=0, and also which direction will be positive. This is referred to as
choosing a coordinate system or choosing a frame of reference. (The two terms
are nearly synonymous, but the first focuses more on the actual x variable,
while the second is more of a general way of referring to one’s point of
view.) As long as you are consistent, any frame is equally valid. You just
don’t want to change coordinate systems in the middle of a calculation.
Have you ever been sitting in a train in a station when suddenly you
notice that the station is moving backward. Most people would describe the
situation by saying that you just failed to notice that the train was moving
— it only seemed like the station was moving. But this shows that there is
yet a third arbitrary choice that goes into choosing a coordinate system:
valid frames of reference can differ from each other by moving relative to
one another. It might seem strange that anyone would bother with a
coordinate system that was moving relative to the earth, but for instance the
frame of reference moving along with a train might be far more convenient
for describing things happening inside the train.
Zero, because the “after” and “before” values of x are the same.
Section 2.2Describing Distance and Time