Ambiguity w/ Newton's dot notation + vector-magnitudes

  • #1
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Stumbled on a potential pitfall that can arise if you combine Newton's overdot notation (for time-derivatives) with boldface/non-boldface notation for vectors/vector-magnitudes.

Say you have ##\textbf{v}## for velocity and ##\dot{\textbf{v}}## for acceleration. Speed is then ##v## (magnitude of velocity vector), but what does ##\dot{v}## mean? There are two possible interpretations:

1) the time-derivative of speed ##v##
2) the magnitude of acceleration ##\dot{\textbf{v}}##

In most contexts there's probably little risk of confusion (#2 can often be safely assumed). But one must be particularly careful in the one-dimensional case, when #2 is the absolute value of #1 and either quantity might prove useful. The danger there is a dreaded sign error.
 

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  • #2
kuruman
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Stumbled on a potential pitfall that can arise if you combine Newton's overdot notation (for time-derivatives) with boldface/non-boldface notation for vectors/vector-magnitudes.

Say you have ##\textbf{v}## for velocity and ##\dot{\textbf{v}}## for acceleration. Speed is then ##v## (magnitude of velocity vector), but what does ##\dot{v}## mean? There are two possible interpretations:

1) the time-derivative of speed ##v##
2) the magnitude of acceleration ##\dot{\textbf{v}}##

In most contexts there's probably little risk of confusion (#2 can often be safely assumed). But one must be particularly careful in the one-dimensional case, when #2 is the absolute value of #1 and either quantity might prove useful. The danger there is a dreaded sign error.
In my opinion one ought to stick to the convention that boldface means "vector" and italics means "scalar". So I would interpret ##\dot v## as the time derivative of the speed. If I wanted to express the magnitude of the acceleration, I would write ##\ddot r## or ##|\dot {\textbf {v}}|## or ##a## . However, your point is well taken as there are people who sometimes carelessly write expressions they don't really mean. The dreaded sign error that you mention is often seen in PF postings where there is vertical projectile motion with air resistance proportional to some power of the speed.

On edit: The correct magnitude of the acceleration should be ##|\ddot {\textbf {r}}|##.
 
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