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Negative velocities in relative motion
My discussion of how to interpret positive and negative signs of velocity
may have left you wondering why we should bother. Why not just make
velocity positive by definition. The original reason why negative numbers
were invented was that bookkeepers decided it would be convenient to use
the negative number concept for payments to distinguish them from
receipts. It was just plain easier than writing receipts in black and payments
in red ink. After adding up your month’s positive receipts and negative
payments, you either got a positive number, indicating profit, or a negative
number, showing a loss. You could then show the that total with a high-
tech “+” or “–” sign, instead of looking around for the appropriate bottle of
ink.
Nowadays we use positive and negative numbers for all kinds of things,
but in every case the point is that it makes sense to add and subtract those
things according to the rules you learned in grade school, such as “minus a
minus makes a plus, why this is true we need not discuss.” Adding velocities
has the significance of comparing relative motion, and with this interpreta-
tion negative and positive velocities can used within a consistent framework.
For example, the truck’s velocity relative to the couch equals the truck’s
velocity relative to the ball plus the ball’s velocity relative to the couch:
v
TC
=
v
TB
+v
BC
= –5 cm/s + 15 cm/s
= 10 cm/s
If we didn’t have the technology of negative numbers, we would have
had to remember a complicated set of rules for adding velocities: (1) if the
two objects are both moving forward, you add, (2) if one is moving forward
and one is moving backward, you subtract, but (3) if they’re both moving
backward, you add. What a pain that would have been.
Discussion questions
A. Interpret the general rule
v
AB
=–v
BA
in words.
B. Wa-Chuen slips away from her father at the mall and walks up the down
escalator, so that she stays in one place. Write this in terms of symbols.
Chapter 2Velocity and Relative Motion
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