- #1
hriby
- 3
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I hope that you've heard about Brachistochrone problem: http://mathworld.wolfram.com/BrachistochroneProblem.html
Given two points, I can find (calculate) the courve, on which the ball needs minimum time to travel from point 1 to point 2.
I get the equation for the courve, which is cycloid, in parametric form, let say:
[tex]
x(\theta)&=&C\left(\theta-\sin{\theta}\right),
[/tex]
[tex]
y(\theta)&=&-C\left(1-\cos{\theta}\right).
[/tex]
Now I also need to calculate the time needed...
How could I calculate it out of formula below using the equation for the courve/cycloid in parametric form?
[tex]t_{12}=\int_{T_1}^{T_2} \frac{\sqrt{1+{y'}^2}}{\sqrt{2g\,y}}dx, \quad y'=\frac{dy}{dx}[/tex]
Thanks for your answers!
Hriby
Given two points, I can find (calculate) the courve, on which the ball needs minimum time to travel from point 1 to point 2.
I get the equation for the courve, which is cycloid, in parametric form, let say:
[tex]
x(\theta)&=&C\left(\theta-\sin{\theta}\right),
[/tex]
[tex]
y(\theta)&=&-C\left(1-\cos{\theta}\right).
[/tex]
Now I also need to calculate the time needed...
How could I calculate it out of formula below using the equation for the courve/cycloid in parametric form?
[tex]t_{12}=\int_{T_1}^{T_2} \frac{\sqrt{1+{y'}^2}}{\sqrt{2g\,y}}dx, \quad y'=\frac{dy}{dx}[/tex]
Thanks for your answers!
Hriby
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