Acceleration as a function of x to a function of time

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  • #1
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IMG_1481241036.236074.jpg
 

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  • #2
BvU
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In 1 dimension ?
 
  • #3
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In 1 dimension ?

Yes
 
  • #4
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And is $$F = -\displaystyle {GMm\over x^2} $$ or is $$F = +\displaystyle{GMm\over x^2} $$ as on the whyteboard ?
 
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And is $$F = -\displaystyle {GMm\over x^2} $$ or is $$F = +\displaystyle{GMm\over x^2} $$ as on the whyteboard ?

Positive
 
  • #8
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I didn't do anything except enter the thing in wolframalpha !
 
  • #9
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I didn't do anything except enter the thing in wolframalpha !

So what does that equation mean?
 
  • #12
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$$\frac{dv}{dt}=-\frac{GM}{x^2}$$If you multiply both sides of this equation by v=dx/dt, you get:$$v\frac{dv}{dt}=-\frac{MG}{x^2}\frac{dx}{dt}$$Both sides of this equation are exact differentials with respect to time.
 
  • #13
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$$\frac{dv}{dt}=-\frac{GM}{x^2}$$If you multiply both sides of this equation by v=dx/dt, you get:$$v\frac{dv}{dt}=-\frac{MG}{x^2}\frac{dx}{dt}$$Both sides of this equation are exact differentials with respect to time.

So is the following correct?

$$v dv = \frac{-MG}{x^2} dx$$
 
  • #15
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Yes.


So how do you integrate over a time interval? That is to say, how do you find the velocity over the interval [0, t]?
 
  • #16
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So how do you integrate over a time interval? That is to say, how do you find the velocity over the interval [0, t]?
Do you know how to solve for v as a function of x?
 

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