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Proving a known function of position via Chain Rule

  • Thread starter kylera
  • Start date
40
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1. Homework Statement
Use the Chain Rule to prove that for rectilinear motion, when the acceleration is a known function of position, you can find the velocity as a function of position via the integral

[tex]\frac{v^{2}-v_{0}^{2}}{2} = \int^{s}_{s_{0}}a(s)ds[/tex]


2. Homework Equations
[tex]v^{2}=v_{0}^{2}\times2as[/tex]


3. The Attempt at a Solution
I took the left fraction, substituted v^2, simplified and got [tex]as[/tex]. I let A be as, then took dA to get a da. Now I'm stuck.
 
250
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I think you simplified wrong. It's a multipication sign not addition. See what I mean?
 
40
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Hold up, I wrote the question on the board wrong -- it is supposed to be a plus for the relevant equation part.
 
38
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I'm not exactly sure what they're asking here. For constant accelerations, you "relevant equation" is basically the answer, assuming you swap out the "x" for a "+" and take a square root. What's throwing me is the request for proof by chain rule.
 

tiny-tim

Science Advisor
Homework Helper
25,790
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Chain Rule

1. Homework Statement
Use the Chain Rule to prove that for rectilinear motion, when the acceleration is a known function of position, you can find the velocity as a function of position via the integral

[tex]\frac{v^{2}-v_{0}^{2}}{2} = \int^{s}_{s_{0}}a(s)ds[/tex]


2. Homework Equations
[tex]v^{2}=v_{0}^{2}\times2as[/tex]


3. The Attempt at a Solution
I took the left fraction, substituted v^2, simplified and got [tex]as[/tex]. I let A be as, then took dA to get a da. Now I'm stuck.
Hi kylera! :smile:

You were asked to use the Chain Rule. So …

Hint: the LHS is ∫vdv. So use the Chain Rule on dv. :smile:
 

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