We saw that the total energy of an orbit (bound or unbound) is given by E=.5mv2-GMm/r, where M is the mass of the parent body, m is the mass of the orbiting body, v is its orbital speed, and r is its distance from the parent body. (a) Use Newton’s second law to show that the total mechanical energy of a circular orbit is E=(-GMm)/(2R), where R is the radius of its orbit. (b) As a general rule, we can replace radius R with semimajor axis a for an elliptical orbit. With this substitution in the above equation, use conservation of energy to derive the vis-viva equation, which relates an orbiting object’s speed v to its current distance from the parent body r: v2=GM((2/r)–(1/a)). (c) Use the vis-viva equation to calculate the orbital speed of the Earth at perihelion and aphelion, given that the semimajor axis of Earth’s orbit is a = 1.496 × 1011 m and the eccentricity of its orbit is e = 0.01671.
The course I am taking is College Physics I.
However, this question was previously submitted and rejected, with a comment that it was an Aerospace Engineering question instead of a Physics question, and I was advised to submit it to an appropriate category. I don't see an Aerospace Engineering category on Bartleby, though; so I am submitting it here again.
Although my Physics professor would probably disagree, if you find, again, that this is not the correct category, please let me know the actual category to which I should submit it. Thank you.
Question/Problem:
We saw that the total energy of an orbit (bound or unbound) is given by E=.5mv2-GMm/r, where M is the mass of the parent body, m is the mass of the orbiting body, v is its orbital speed, and r is its distance from the parent body. (a) Use Newton’s second law to show that the total mechanical energy of a circular orbit is E=(-GMm)/(2R), where R is the radius of its orbit. (b) As a general rule, we can replace radius R with semimajor axis a for an elliptical orbit. With this substitution in the above equation, use conservation of energy to derive the vis-viva equation, which relates an orbiting object’s speed v to its current distance from the parent body r: v2=GM((2/r)–(1/a)). (c) Use the vis-viva equation to calculate the orbital speed of the Earth at perihelion and aphelion, given that the semimajor axis of Earth’s orbit is a = 1.496 × 1011 m and the eccentricity of its orbit is e = 0.01671.
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