What is the eccentricity of an orbit such that Vp = 2Va?

In summary: Maybe the professor missed the square on the right hand side of$$\frac{v_p^2}{v_a^2} = \left(\frac{1+e}{1-e}\right)^2$$In summary, the professor is providing an answer that is based on an equation that may not be correct.
  • #1
csmiller1993
2
1
Summary:: A question on a recent exam was, "At what eccentricity does an orbit experience a velocity at periapsis that is twice the velocity an apoapsis?" I don't know why the provided solution is correct.

On a recent exam, one of the questions was "At what eccentricity does an orbit experience a velocity at periapsis that is twice the velocity an apoapsis?"

In the exam solutions, the given answer is e = 3/5. with the following as the justification:

e + 1 = 4 - 4e
e = 3/5

I'm not sure where the professor got this relation. When I attempt to solve this problem I end up with e = 1/3. My work is attached. Am I missing something or is the professor wrong? I emailed her and she simply said to check my derivation. I've solved it with the vis viva equation as well and still came out to 1/3. Other students have also reported that they got 1/3.

Thanks in advance!
 

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  • #2
I did it using ##e=\sqrt{1-\frac{b^2}{a^2}}## and got the same answer as you. You have the right to learn from your teacher how to do it correctly. I think it's time to show her your solution and say you couldn't find anything wrong with it. Then ask her what equation she used. It's entirely possible that she made a calculational mistake which she will find out once you ask. You will be doing yourself and your classmates a favor.
 
  • #3
I did it using ##e=\sqrt{1-\frac{b^2}{a^2}}## and got the same answer as you. I think it's time to show her your solution and couldn't find anything wrong then ask her what formula she used. It's entirely possible that she made a mistake. As her student, you have the right to learn from her how to do it correctly.
 
  • #4
kuruman said:
I did it using ##e=\sqrt{1-\frac{b^2}{a^2}}## and got the same answer as you. I think it's time to show her your solution and couldn't find anything wrong then ask her what formula she used. It's entirely possible that she made a mistake. As her student, you have the right to learn from her how to do it correctly.
Thanks for the reply and for solving it through for yourself. I think I will talk to her because you're right, there's no reason other's should be losing points over this or be confused by it.
 
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  • #5
I would give twice full credit to the first student who pointed a mistake like this to me on a test. It didn't happen often but encouraged students to speak up if they perceived something wrong.
 
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  • #6
I also get ##e=1/3##. Think perhaps the professor missed the square on the right hand side of
$$\frac{v_p^2}{v_a^2} = \left(\frac{1+e}{1-e}\right)^2$$
 
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Related to What is the eccentricity of an orbit such that Vp = 2Va?

1. What is the definition of eccentricity in orbital mechanics?

Eccentricity is a measure of how elliptical an orbit is, with a value of 0 representing a perfect circle and a value of 1 representing a parabolic orbit.

2. How is eccentricity related to the velocity of an orbiting object?

The eccentricity of an orbit determines the ratio of the object's closest approach to its farthest distance from the body it is orbiting. This ratio, along with the distance between the two objects, determines the object's velocity at any given point in its orbit.

3. What does it mean for Vp to equal 2Va in terms of eccentricity?

When Vp (velocity at perihelion) is equal to 2Va (velocity at aphelion), the eccentricity of the orbit is 1. This means that the orbit is a parabola and the object has enough velocity to escape the gravitational pull of the body it is orbiting.

4. How does the eccentricity of an orbit affect the shape of the orbit?

The eccentricity of an orbit determines the shape of the orbit, with higher eccentricities resulting in more elongated and elliptical orbits. A lower eccentricity results in a more circular orbit.

5. How is eccentricity calculated for an orbit?

Eccentricity can be calculated by dividing the distance between the foci of the ellipse (the two objects being orbited) by the length of the major axis of the ellipse. This value can also be determined using the object's closest and farthest distances from the body it is orbiting.

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