Is there a more rigorous way to prove this?

  • Thread starter Keshroom
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  • #1
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Homework Statement


Show that d(v^2)/dt = 2 . (d^2r/dt^2) . (dr/dt)

HINT: v^2 = ||dr/dt||^2 = dr/dt . dr/dt

Homework Equations





The Attempt at a Solution



I did it another method:

d(v^2)/dt = d/dv(v^2) . dv/dt --------------chain rule
= 2v . dv/dt
since v=dr/dt and dv/dt = d^2r/dt^2
therefore = 2 . (d^2r/dt^2) . (dr/dt)

Is there a more rigorous method to solving this, using the 'HINT' it gave me. I can't quite figure it out.
 
Last edited:

Answers and Replies

  • #2
tiny-tim
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Hi Keshroom! :smile:

(try using the X2 button just above the Reply box :wink:)

Sorry, but this doesn't work at all. :redface:
Show that d(v^2)/dt = 2 . (d^2/dt^2) . (dr/dt)

HINT: v^2 = ||dr/dt||^2 = dr/dt . dr/dt

I did it another method:

d(v^2)/dt = d/dv(v^2) . dv/dt --------------chain rule
= 2v . dv/dt
since v=dr/dt and dv/dt = dr^2/dt^2

(your "." is ordinary multiplication of two scalars, unlike the question, where it's the dot-product of two vectors; but anyway:)

no, dv/dt is not d2r/dt2 :redface:

(except in one-dimensional motion)
Is there a more rigorous method to solving this, using the 'HINT' it gave me.

it's easier if you rewrite the hint as:

v2 = r' . r' :wink:
 
  • #3
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Hi Keshroom! :smile:

(try using the X2 button just above the Reply box :wink:)

Sorry, but this doesn't work at all. :redface:


(your "." is ordinary multiplication of two scalars, unlike the question, where it's the dot-product of two vectors; but anyway:)

no, dv/dt is not d2r/dt2 :redface:

(except in one-dimensional motion)


it's easier if you rewrite the hint as:

v2 = r' . r' :wink:
Sorry i'm still learning with the notation and everything. Awww ok. You mind telling me how to prove this please?
thanks

btw there were errors in the question which i have edited now
 
Last edited:
  • #4
tiny-tim
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  • #5
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just differentiate r' . r' using the product rule :smile:

lol what....so simple. So it's just
r'r'' + r'r'' = 2r'r'' ?
 
Last edited:
  • #6
tiny-tim
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no, it's r'.r'' + r'.r'' = 2r'.r'' :wink:
 
  • #7
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--_--

haha thanks!
 

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