Understanding the Vis-Viva Equation for Elliptical Orbits

[fargoth]

im a private teacher now for first year students... and today my student asked me a question i didnt know the answer to... so i said i'll look it up, but I am going out with friends :tongue2:

so if you could help and save me the time id really appreciate it.

the problem is finding the maximum radius of an eliptic route.
we know the tangential speed of the body at the point of the minimal radius.
so according to the advertised solution for the problem, the equation should be:
$$\frac{mv_{max}^2}{2}-\frac{A}{R_{min}}=\frac{(mv_{max}R_{min})^2}{2mR_{max}^2}-\frac{A}{R_{max}}$$
but we couldn't understand why it was $$E_k+U=U_{eff}$$ and not $$E_k+U_{eff}(R_{min})=U_{eff}(R_{max})$$

 

[fargoth]

nm, found my answer... while thinking at the pub

 

[kyle8921]

Thank you for reaching out for assistance with this problem. I am always happy to help others learn and understand scientific concepts. It sounds like you are doing a great job as a private teacher for first year students.

To answer your question, the equation you have provided is known as the vis-viva equation, which relates the kinetic and potential energies of a body in orbit. The equation is typically written as E = \frac{mv^2}{2} - \frac{GmM}{r}, where E is the total energy, m is the mass of the orbiting body, v is the tangential velocity, G is the gravitational constant, M is the mass of the central body, and r is the distance between the two bodies.

In this case, the advertised solution is using the specific form of the vis-viva equation for elliptical orbits, where A is the semi-major axis of the orbit and R is the distance between the orbiting body and the central body. The first term in the equation represents the kinetic energy of the orbiting body at its maximum velocity, while the second term represents the potential energy at the minimum radius of the orbit. The third and fourth terms represent the kinetic and potential energies at the maximum radius of the orbit.

As for the confusion regarding E_k+U=U_{eff} and E_k+U_{eff}(R_{min})=U_{eff}(R_{max}), it is important to note that the effective potential energy, U_{eff}, is a function of the distance between the orbiting body and the central body. This means that the potential energy at the minimum radius, U_{eff}(R_{min}), is not the same as the potential energy at the maximum radius, U_{eff}(R_{max}). Therefore, the correct equation should be E_k+U_{eff}(R_{min})=U_{eff}(R_{max}). I hope this clears up any confusion and helps you understand the problem better. Good luck with your teaching and studies!