Orbiting Ranking

Classical Mechanics Level 2

Consider the two orbits around the sun shown below. Orbit $P$ is circular with radius $R$ , orbit $Q$ is elliptical such that the farthest point b is between $2R$ and $3R$ , and the nearest point a is between $R/3$ and $R/2$ . Consider the magnitudes of the velocity of the circular orbit $v_c$ , the velocity of the comet in the elliptical orbit at the farthest point $v_b$ , and the velocity of the comet in the elliptical orbit at the nearest point $v_a$ . Which of the following rankings is correct?

Note : This problem appeared on the 2012 F = ma exam.

2v_c > v_b > v_a

10v_b > v_a > v_c

v_b > v_c > 2 v_a

v_c > v_a > 4v_b

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Anand Raj
Feb 7, 2014

We Can See that Va > Vc > Vb By looking into the diagram.........

This rules out all the three except the answer.

Jake Maason
Jun 12, 2014

(I did it the hard way for fun) First I just pretended that orbit P was Earth around the sun, that automatically gives us R = 1.496x10^11 and vc = 30,000 m/s. Then I find the midpoint values between 2R and 3R, I do the same with R/2 and R3 to get the closest and furthest points for orbit Q.

After that, it's just a simple application of finding the orbital velocity for point va and vb using sqrt(GM/r). So we find that va is about 46000 m/s vb is about 20000 m/s and vc is still 30000 m/s

Then looking at the possible answers, it is clear that only one of them makes sense. Alternatively, we could have simply observed that va > vc > vb and could have assigned them easy to work with values such as 3,2,1.

Kshitij Khandelwal
Mar 12, 2014

angular momentum will be conserved L=mvr...
so Va>Vc>Vb now see options. but solution is for maximum Va after fixing Vb, Vb (3R) = Va (R/3) that gives maximum Va=9Vb

Shaik Asif Ahmad
Feb 12, 2014

This problem makes use of Kepler's Third Law which states that the farther the orbit,the lesser the orbital speed with which the object is revolving around the sun So its obvious that Va>Vc>Vb I solved this problem by eliminating the other options,which could not be true taking into account the above inequality. Only option A satisfied the problem and thats it guys!!

Orbiting Ranking

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