Elliptical Orbits velocity and distance

[garva]

This question is of an elliptical orbit with v1 ant its distance r1 and a second v2 with distance r2 located a different point in the elliptical orbit
Which general relation connects the speed
vi of the satellite and its distance ri from
the center of mass of the planet at any two
arbitrary points on the orbit?

I thought it was r1 v1 = r2 v2 based on Keplers 2 law
but that is not the answer, and I am confused between these two i know it might be a because
that is the equation of total energy, but what confuse me is that its multiplied by 2

a)v^2−(2GM/r1)= v^2−(2GM/r2)

b)v^2/r1=v^2/r2

I hope you all can clear my confusion thank you

 

[D H]

I thought it was r1 v1 = r2 v2 based on Keplers 2 law

One way to express Kepler's 2nd law is that $r v_{\perp} = \text{constant}$, where $r v_{\perp}$ is the component of the velocity perpendicular to the line connecting the center of the planet and the satellite.

im confused between these two i know it might be a because
that is the equation of total energy, but what confuse me is that its multiplied by 2

a)v^2−(2GM/r1)= v^2−(2GM/r2)

b)v^2/r1=v^2/r2

The total energy of the satellite doesn't change; it is a constant of the orbit. If you multiply a constant by 2 (or by any other constant) you still have a constant.

One of those equations is indeed just the total energy times some constant. The other one is wrong.

 

[tomcue]

The correct answer is b) v^2/r1 = v^2/r2. This is known as the conservation of angular momentum and is one of the fundamental laws of orbital mechanics. This law states that the product of the satellite's speed and its distance from the center of mass remains constant at any point on the orbit. This is also known as the "area law" and is a consequence of Kepler's Second Law.

The equation you mentioned, r1v1 = r2v2, is known as the "vis-viva equation" and is used to calculate the velocity of a satellite at any point on its orbit. It is derived from the conservation of energy and is not directly related to the conservation of angular momentum.

The reason for the confusion may be because both equations involve the variables r1 and r2, but they represent different physical quantities. The first equation represents the conservation of energy, while the second equation represents the conservation of angular momentum. Both are important in understanding the motion of objects in elliptical orbits.

I hope this helps to clear up your confusion. It is important to understand the fundamental laws and equations of orbital mechanics in order to accurately study and predict the behavior of objects in space.