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How can I calculate the time taken to go from one point to another, in vertical circular motion? If we have radius, angle between 2 points, and initial velocity.

I tried to write $\frac{dv}{dt} = g \cos \left(\frac{vt}{r}\right)$, but I don't know how to integrate that. I also tried to write velocity as a function of $\theta$ using conservation of energy, but again was not able to get anything out of it.

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    $\begingroup$ Hi and welcome to the Physics SE! Please note that this is not a homework help site. Please see this Meta post on asking homework questions and this Meta post for "check my work" problems. $\endgroup$
    – John Rennie
    Commented Oct 12, 2015 at 11:15
  • $\begingroup$ Hey sorry I don't really understand sometimes if the question is a doubt or a homework question. I mean, I post here with the intention to learn something, not at all to get an answer. Besides, I don't get any homework as such, so they're all my own doubts. Thanks for your concern though! $\endgroup$
    – Shodai
    Commented Oct 12, 2015 at 13:39

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By conservation of energy, if initial velocity is $v$ then the velocity at an angle $a$ is given by $u=\sqrt{v^2 -2gR(1-\cos a)}.$ Now let the object move through a distance $ds.$ $ds=R\cdot da$ ($da$=small angle subtended by $ds$ at centre). Use $dt=\frac{ds}{v}.$ Integrate RHS from angle $0$ to $a$ and similarly integrate LHS from time $0$ to $t.$ Now substitute value in the equation to get time $t.$

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  • $\begingroup$ Yes, I did that, but how to integrate ds/v? I was unable to do that. $\endgroup$
    – Shodai
    Commented Oct 12, 2015 at 13:38

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