Show that a curve has constant speed if and only if its acceleration is everywhere orthogonal to its velocity.

Shannon Andrews 2022-07-25 Answered
Show that a curve has constant speed if and only if its acceleration is everywhere orthogonal to its velocity.
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Answers (1)

Cheyanne Charles
Answered 2022-07-26 Author has 13 answers
Let the curve be r(t). Then, the velocity of the curve is v(r) =r'(t).
Use the deivative of the product,
d d t ( ν ( t ) , ν ( t ) ) = ν ( t ) ν ( t ) + ν ( t ) ν ( t ) = 2 ν ( t ) ν ( t ) = 2 ν ( t ) a ( t )
Necessary part:
Let the curve has a constant speed. That is, ν ( t ) ∥= c . To prove, acceleration is everywhere
orthogonal to its velocity.
ν ( t ) ∥= c d d t ( r ( t ) ) 2 = 0 d d t ( ν ( t ) , ν ( t ) ) = 0 2 ν ( t ) a ( t ) = 0 ν ( t ) a ( t ) = 0Copyright ©2011-2012 CUI WEI. All Rights Reserved.
Hence, acceleration is everywhere orthogonal to its velocity.
Sufficient part:
ν ( t ) a ( t ) = 0 2 ν ( t ) a ( t ) = 0 d d t ( ν ( t ) ν ( t ) ) = 0 ν ( t ) ∥= ν ( t ) , ν ( t ) constant
Hence the proof.
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