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Old Oct 30th 2016, 04:00 PM   #1
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Kepler's Laws vs Energy

Screenshot by Lightshot

The question is in the link above.
I have already looked at unofficial markscheme for this, and it appears to have used vr=v'r' as a form of kepplers law of equal areas, which results in C being circled.
I used a different method, to get B, where I equated the PE and KE to get v^2 = k/r, and then solved for k. Then I plugged it back in with the new radius to get a new velocity of about 30. I have no clue whether or not my method holds, nor why, so it'd be nice to get some clarification on this.

Thanks
Btw im new here, apologies if this isnt where it should be

Last edited by haksaw22; Oct 30th 2016 at 05:55 PM.
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Old Oct 30th 2016, 05:08 PM   #2
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Yes, the answer to the question as written is "about" 30 - actually 31.6Km/s.

I'm wondering if there isn't a typo in tjhe question - if the radius of the orbit at maximum was 10 x 10^11Km instead of 10 x 10^10Km, there would be a nice round number for the answer.
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Last edited by ChipB; Oct 30th 2016 at 05:17 PM.
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Old Oct 30th 2016, 05:39 PM   #3
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Thanks for the confirmation! I'm pretty sure there isn't a typo, which is weird given that its a non calc paper. Do you know why the law of equal areas method they've used doesn't work?
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Old Oct 30th 2016, 10:12 PM   #4
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Sorry, but I realize I made a mistake in my earlier post. Using Kepler's Law of equal area in equal time does indeed yield an answer of (c), which is 20 Km/s. My error (and I think yours too) was in mistakenly using an equation that works for circular orbits, but this problem is about an elliptical orbit. I used a formula that equates force of gravity to mass times centripetal acceleration of the comet, which is valid for a circular orbit:

GMm/r^2 = m v^2/r

This yields v^2r = constant, but ignores the fact that at perigee and apogee the comet in an elliptical orbit has radial as well as centripetal acceleration, so gives the wrong answer.

Sorry for any confusion on this.
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Last edited by ChipB; Oct 31st 2016 at 07:14 AM.
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Old Oct 31st 2016, 04:20 AM   #5
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No problem, thanks again! Makes more sense now, subtleties always catch me out.
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