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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