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sanitykey
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[SOLVED] Free fall far away from Earth (integral substitution problem)
Given:
[tex]v(x) = -v_1\sqrt{\left(\frac{R}{x} - \frac{R}{h}\right)}[/tex]
Find the time t.
Listed above where [tex]v_1 , R , h[/tex] are all constant.
[tex]v(x) = \frac{dx}{dt}[/tex]
[tex]dt = \frac{dx}{\left[-v_1\sqrt{\left(\frac{R}{x} - \frac{R}{h}\right)}\right]}[/tex]
[tex]t = -\frac{1}{v_1}\int\frac{dx}{\left[\sqrt{\left(\frac{R}{x} - \frac{R}{h}\right)}\right]}[/tex]
I'm stuck on this integral but I'm convinced it's some kind of trigonometric substitution although i can't figure out which one. I've spent hours searching the internet for help and couldn't find anything so any help would be appreciated.
Perhaps if someone could at least point out which trigonometric substitution (if any at all) i should be trying to work with i'd be extremely grateful.
Thanks in advance.
Homework Statement
Given:
[tex]v(x) = -v_1\sqrt{\left(\frac{R}{x} - \frac{R}{h}\right)}[/tex]
Find the time t.
Homework Equations
Listed above where [tex]v_1 , R , h[/tex] are all constant.
The Attempt at a Solution
[tex]v(x) = \frac{dx}{dt}[/tex]
[tex]dt = \frac{dx}{\left[-v_1\sqrt{\left(\frac{R}{x} - \frac{R}{h}\right)}\right]}[/tex]
[tex]t = -\frac{1}{v_1}\int\frac{dx}{\left[\sqrt{\left(\frac{R}{x} - \frac{R}{h}\right)}\right]}[/tex]
I'm stuck on this integral but I'm convinced it's some kind of trigonometric substitution although i can't figure out which one. I've spent hours searching the internet for help and couldn't find anything so any help would be appreciated.
Perhaps if someone could at least point out which trigonometric substitution (if any at all) i should be trying to work with i'd be extremely grateful.
Thanks in advance.