Greetings,(adsbygoogle = window.adsbygoogle || []).push({});

I am given the following equation.

[tex]t-t_{0}=\int_{R_{0}}^{R(t)}\frac{du}{\sqrt{2mEL^{-2}u^4-u^2-2mL^{-2}u^4V(u)}}[/tex]

This is the total energy of a prticle moving in a central conservative field. m = mass, E = energy, L = angular momentum. The force the particle experiences is F = -Hmu^-3, where H is some constant, m is the mass of the particle, and u the distance. V(u), the potential, is just the negative integral of the force, and it is

[tex]-Hm/2u^2[/tex]

How can I show that the energy, E, if E < 0, then

[tex]\alpha(t-t_{0})=\int_{R_{0}}^{R(\Theta)}\frac{du}{u\sqrt{a^2-u^2}} [/tex]

, for real numbers alpha and a.

And similarly, for E > 0, how can I show it's

[tex]\beta(t-t_{0})=\int_{R_{0}}^{R(\Theta)}\frac{du}{u\sqrt{u^2+b}} [/tex]

, for some real numbers beta and b?

I really need help on this!

I did show, for the E = 0 case, how it should be done. Please take the time check my work for this part.

Since E = 0, the total energy equation simplifies to:

[tex]t-t_{0}=\int_{R_{0}}^{R(\Theta)}\frac{du}{\sqrt{-u^2-2mL^{-2}u^4V(u)}}[/tex]

then, plugging in V(u),

[tex]t-t_{0}=\int_{R_{0}}^{R(\Theta)}\frac{du}{\sqrt{-u^2+m^2L^{-2}Hu^2}}[/tex]

[tex] Let s = m^2L^{-2}H[/tex]

[tex]t-t_{0}=\int_{R_{0}}^{R(\Theta)}\frac{du}{\sqrt{-u^2+su^2}}[/tex]

[tex]t-t_{0}=\int_{R_{0}}^{R(\Theta)}\frac{du}{u\sqrt{-1+s}}[/tex]

[tex]t-t_{0}=\int_{R_{0}}^{R(\Theta)}\frac{du}{u\sqrt{-1+s}}[/tex]

Since s is only a bunch of constants, we can factor it out.

[tex]t-t_{0}=\frac{1}{\sqrt{-1+s}}\int_{R_{0}}^{R(t)}\frac{du}{u}[/tex]

[tex] \ln {R_{\Theta}/R_{0}} = \sqrt{s-1)[/tex]

then solve for [tex] R (\Theta) [/tex]

[tex] R (\Theta) = R_{0}e^{(\Theta-\Theta_{0})\sqrt{s-1}}[/tex]

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Integration/Proving help! And check my work please!

**Physics Forums | Science Articles, Homework Help, Discussion**