|Apr8-11, 02:50 PM||#1|
planetary orbit energies
1. The problem statement, all variables and given/known data
A satellite is to be launched from earth to venus. It is to make two complete circuits of the earth, then travel to venus, and complete one orbit of venus in 12 hours. Find the velocities of the satellite around venus, around the earth, and the radius of the orbit around Venus. I attempted to help a high school student with this question, and think that certain necessary data is missing; furthermore the student does not know (and has not been taught, nor will be taught during this course) the equation for gravitational potential energy in an orbit. My impression is that the question cannot be answered without more givens. Am I right? Or am I missing something fundamental?
2. Relevant equations
Around venus: v = 2 pi R/T, centripetal force = m v^2/R, Gravitational force = G m M[SUB]v[SUB]/R^2, (KE + grav PE) at Venus orbit = (KE + PE) at earth orbit. Mass of Venus = 4.87 E24 kg, etc. The equation for gravitational potential energy, U = -G m M/R is not available to the student.
3. The attempt at a solution
For the venus orbit: I converted T from hours to seconds, set centripetal force equal to orbital graviational force, and found that the orbital radius is 2.49 E 7 meters and orbital velocity is 3.62 km/s. So far, quite simple. Using the equation for gravitational potential energy at the calculated orbital radius, I could calculate the gravitational potential energy of the satellite around Venus; however, as mentioned above, the student does not have the use of the equation U = -G m M/R. (I used to teach this class from the same text that the student is using--the equation is not given in the text, and the student, a 9th grader, has not learned calculus so as to derive it from the force equation.)
So, at this point, I'm stuck. Since nothing was given about the orbital distance from earth, nor about its orbital period, I can't calculate the satellite's energy in earth orbit. Since I don't know the potential energy in the Venus orbit, I can't calculate the total energy available in the earth orbit.
Another approach I thought of was to calculate the speed with which the satellite would have to leave earth so as to arrive at Venus with the correct calculated speed. In this case the change in potential energy is due to work done by the gravitational field--but since the force depends on distance, acceleration is not constant, so again the problem is too advanced for the student.
So, in conclusion, I'm asking, is my reasoning correct? Is the problem statement missing something? Or am I missing something? Thanks in advance for feedback.
|Apr8-11, 04:59 PM||#2|
The problem is definitely either missing information, or unsolvable. Note that the flight trajectory it describes is impossible unless the spacecraft's engines fire. If they don't, spacecraft in closed orbits always stay in closed orbits, and spacecraft in escape trajectories always stay in escape trajectories. There is no way to complete one orbit of Earth without completing an infinite number of orbits, and no way to get to Venus without either coming back to the point of origin or escaping Venus' orbit.
If the engines do fire, the spacecraft could have any velocity at all around the Earth. After all, the engines will have to change this velocity to get to Venus, so it doesn't matter what the velocity was initially.
Could you ask the student to post the question here himself?
|Apr8-11, 05:44 PM||#3|
Thanks--I'll ask the student to post the question herself the next time I see her--Sara
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