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

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Cheyanne Charles · Accepted Answer

Let the curve be r(t). Then, the velocity of the curve is v(r) =r&#039;(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 everywhereorthogonal 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    )  constantHence the proof.

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

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