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Mechanics}Orbits

The planet orbits in an ellipse, and thus the sum of the minimum and maximum distance is twice the semimajor axis of the orbit. Choice (E) is out.

From Kepler's third law (which can be derived from the formalism of central force theory), one has T^2\propto r^3. Thus, knowing the distances, one can find the period. Choice (D) is out.

The speed can be found from conservation of energy. For an elliptical orbit, one has E=-k/(2a) = 1/2 mv^2 - k/r, where k=GmM and a is the semimajor axis. (One can find the total energy of the orbit from the Virial Theorem. Recall that \langle T \rangle = -\langle \sum F\cdot r\rangle/2. Plug in the gravitational force F=k/r^2 to get \langle T \rangle = \langle k/2r \rangle = \langle k/2a \rangle. Thus, T+V = k/2a - k/a = -k/2a.)

The full form of Kepler's third law is \tau^2 \approx a^3/(GM). Thus, one can determine the mass of the planet.

The only one that remains is choice (A).
See below for user comments and alternate solutions! See below for user comments and alternate solutions!
Alternate Solutions
amber
2014-10-22 17:33:43		I just remembered SPUTNIK.

The Soviet Union launched it and we couldn't figure out its mass.		Alternate Solution - Unverified
Ning Bao
2008-02-01 06:11:15		Consider the asteroid belt. They're not all the same size. Thus, their orbits are not determined by masses, because certainly in the muck there are a few asteroids of wildly varying size with very similar orbits.Alternate Solution - Unverified
KarstenChu
2007-03-22 14:03:05		It's possible, it seems, to just look at the case of a circular orbit. Setting GMm/r^2=mv^2/r reveals that m cancels out....no mass of the moon for you!Alternate Solution - Unverified
travis.nicholson
2006-10-23 01:49:09		A more appropriate solution would be:

2a = r_{min} + r_{max}, where a is the semimajor axis. This elimiates choice (E).

The angular momentum of the moon is conserved because there are no external torques. Therefore l_{min} = l_{max} -> m v_{min} r_{min} = m v_{max} r_{max} -> v_{min} = {v_{max} r_{max}}/r_{min}, where l is the magnitude of the angular momentum, and m is the mass of the moon. This eliminates choice (C).

Since there is no external work being done on or by the moon, the mechanical energy of the moon is conserved. Also, since the potential energy of the moon goes as 1/r^2, the potential energy is minimal at r_{max} and maximal at r_{min}. Thus, the kinetic energy (and therefore the velocity) is maximal at r_{max} and minimal at r_{min}. With the v_{min} now known (as explained in the previous paragraph), the mass of the planet, M, can be calculated using the conservation of energy:

{m {v_{max}}^2}/2 + {G m M}/{r_{min}}^2 = {m {v_{min}}^2}/2 + {G m M}/{r_{max}}^2. Note again that m cancels. This eliminates choice (B).

Now that the mass of the planet is known, the period, \tau, can be calculated using Kepler's third law:

\tau^2 = {4 \pi^2 a^3}/{G M}. This eliminates choice (D).

Only choice (A) remains.		Alternate Solution - Unverified
travis.nicholson
2006-10-23 01:40:58		A more appropriate solution would be:

2a = r_min + r_max, where a is the semimajor axis. This elimiates choice (E).

The angular momentum of the moon is conserved because there are no external torques. Therefore l_min = l_max -> m v_min r_min = m v_max r_max -> v_min = {v_max r_max}/r_min, where l is the magnitude of the angular momentum, and m is the mass of the moon. This eliminates choice (C).

Since there is no external work being done on or by the moon, the mechanical energy of the moon is conserved. Also, since the potential energy of the moon goes as 1/r^2, the potential energy is minimal at r_max and maximal at r_min. Thus, the kinetic energy (and therefore the velocity) is maximal at r_max and minimal at r_min. With the v_min now known (as explained in the previous paragraph), the mass of the planet, M, can be calculated using the conservation of energy:

{m {v_max}^2}/2 + {G m M}/{r_min}^2 = {m {v_min}^2}/2 + {G m M}/{r_max}^2. Note again that m cancels. This eliminates choice (B).

Now that the mass of the planet is known, the period, \tau, can be calculated using Kepler's third law:

\tau^2 = {4 \pi^2 a^3}/{G M}. This eliminates choice (D).

Only choice (A) remains.
travis.nicholson
2006-10-23 01:50:57		 Sorry ... for some reason when I attempted to fix the syntax in this solution the page would not allow me to submit the edited version. Therefore, I posted the solution again.
Alternate Solution - Unverified
Comments
amber
2014-10-22 17:33:43		I just remembered SPUTNIK.

The Soviet Union launched it and we couldn't figure out its mass.
NervousWreck
2017-03-28 10:25:53		 Same mass as for the nuclear warhead man, same mass.
Alternate Solution - Unverified
solar39
2009-11-04 09:52:25		When you try solving the period with this equation you're actually solving it for this Lagrangian : rn\ L=\frac{1}{2}\m_r\dot{r}^2+\frac{GM_pm_m}{r}rnwhere \m_r is the reduced mass of the planet and the moon.rnThe Euler-Lagrange equations would give you the equations of motion as: \m_r\ddot{r}-\frac{GM_pm_m}{r^2}=0rnand the energy conservation equations change as:rn\frac{1}{2}\m_r\dot{r}^2+\frac{GM_pm_m}{r}=Ernso i guess you do have the "summation" to solve for the period but you can't get \ M_p and\ m_m seperately.rnI think ETS would have wanted you to just think of this problem as a "Kepler problem in approximate form."NEC
mlg52
2008-09-27 23:01:00		 I take issue with this answer! This is the way I did it.

1. Given Rmin and Rmax you can sum them and divide in two to determine the semi-major axis, a. Thus choice E is eliminated

2. Given Rmin and the semi-major axis you just found, you can compute the eccentricity:
Rmin=a(1-e)
where a is the semi-major axis and e is the eccentricity

3. Given Vperi and Rmin you can calculate the mass of the planet:
Vperi = sqrt{(GM(1+e))/Rmin}
Thus choice B is eliminated

4. Similarly, since Rmax was also measured, you can now find the minimum velocity, which will occur at apastron:
Vap= sqrt{(GM(1-e))/Rmax

Thus choice C is eliminated

Now we are left with the mass of the moon or the period of the orbit. Kepler's law is *actually*

P^2= (4pi^2)a^3/(G(M+m))

Note the denominator contains the *sum* of the masses!

So we truly have one equation left with two unknowns. You *cannot* calculate the *exact* period from here. You can only make an approximation where you neglect the mass of the moon, but that seems like a self-fulfilling prophecy or circular reasoning to me. Saying "a very small moon" is very subjective. What if the planet, too, is very small? They don't state the ratio of the planet to the moon, or mention that it is negligible in size compared to the planet. So how can we really use Kepler's law in approximate form in this case?		Help
mlg52
2008-09-27 22:58:20		I take issue with this answer! This is the way I did it.

1. Given Rmin and Rmax you can sum them and divide in two to determine the semi-major axis, a. Thus choice E is eliminated

2. Given Rmin and the semi-major axis you just found, you can compute the eccentricity:
Rmin=a(1-e)
where a is the semi-major axis and e is the eccentricity

3. Given Vperi and Rmin you can calculate the mass of the planet:
Vperi = sqrt{(GM(1+e))/Rmin}
Thus choice B is eliminated

4. Similarly, since Rmax was also measured, you can now find the minimum velocity, which will occur at apastron:
Vap= sqrt{(GM(1-e))/Rmax

Thus choice C is eliminated

Now we are left with the mass of the moon or the period of the orbit. Kepler's law is *actually*

P^2= (4pi^2)a^3/(G(M+m))

Note the denominator contains the *sum* of the masses!

So we truly have one equation left with two unknowns. You *cannot* calculate the *exact* period from here. You can only make an approximation where you neglect the mass of the moon, but that seems like a self-fulfilling prophecy or circular reasoning to me. Saying "a very small moon" is very subjective. What if the planet, too, is very small? They don't state the ratio of the planet to the moon, or mention that it is negligible in size compared to the planet. So how can we really use Kepler's law in approximate form in this case?
NEC
Ning Bao
2008-02-01 06:11:15		Consider the asteroid belt. They're not all the same size. Thus, their orbits are not determined by masses, because certainly in the muck there are a few asteroids of wildly varying size with very similar orbits.Alternate Solution - Unverified
bootstrap
2007-04-06 19:17:26		There is a very easy approach to this. It involves no math, just a basic understanding of the concept. Lets imagine that its not a "very small moon" and a planet, but a satilite orbiting Earth. It could even be a random rock that got stuck into orbit. Both can have the exact same measurements that are provided by this question. So you cant you tell the difference between a rock and a satilite. Another important think to note is the difference between the mass of the earth and a satilite. Maybe for a heavy object about 10000Kg compared to the Earth or some other planet of the order 10^24Kg. If they didnt use the words "very small" then the mass of the moon is important, and you are able to find it (due to other effect that im not going to get into, b/c i believe you need a computer or a day to do the calculations).NEC
KarstenChu
2007-03-22 14:03:05		It's possible, it seems, to just look at the case of a circular orbit. Setting GMm/r^2=mv^2/r reveals that m cancels out....no mass of the moon for you!
redmomatt
2011-10-04 11:59:18		 This is how I did it as well.

We know from Kepler's Law that (D) and (E) both can't be false; knowing one gives you the other.

Likewise, we know the minimum and maximum distances (the effective r value) where, at both points in the orbit, we either also know v and/or M.

Since, by assuming circular orbit F=\frac{GMm}{r^2}=\frac{mv^2}{r} we can either figure out M knowing v or v knowing r, but m always cancels.

This leaves only (A).
Alternate Solution - Unverified
merdeonme
2007-03-11 02:35:06		two corrections:



you have:



L_{min} = L_{max} \Rightarrow mv_{min}r_{min} = mv_{max}r_{max} \Rightarrow v_{min} = \frac{mv_{max}r_{max}}{r_{min}}



When it should be:



L_{min} = L_{max} \Rightarrow mv_{min}r_{max} = mv_{max}r_{min} \Rightarrow v_{min} = \frac{v_{max}r_{min}}{r_{max}}



You also have:



\frac{1}{2}mv_{max}^2 + \frac{GmM}{r_{min}^2} = \frac{1}{2}mv_{min}^2 + \frac{GmM}{r_{max}^2}



when it should be:



\frac{1}{2}mv_{max}^2 + \frac{GmM}{r_{min}} = \frac{1}{2}mv_{min}^2 + \frac{GmM}{r_{max}}
merdeonme
2007-03-11 02:35:52		 Meant this to be a reply to travis.nicholson
NEC
michealmas
2007-01-16 15:27:22		Fools. The planet's mass is overwhelming, so the orbiting moon moves exactly like a brick or pen. They all fall (orbit) at the same rate. Hence, A.
mhas035
2007-04-07 18:06:30		 It seems that you are the one that is a fool, and if you can't figure out why, that only confirms your extreme foolishness.
grae313
2007-11-01 15:40:25		 michealmas is a jackass
NEC
travis.nicholson
2006-10-23 01:49:09		A more appropriate solution would be:

2a = r_{min} + r_{max}, where a is the semimajor axis. This elimiates choice (E).

The angular momentum of the moon is conserved because there are no external torques. Therefore l_{min} = l_{max} -> m v_{min} r_{min} = m v_{max} r_{max} -> v_{min} = {v_{max} r_{max}}/r_{min}, where l is the magnitude of the angular momentum, and m is the mass of the moon. This eliminates choice (C).

Since there is no external work being done on or by the moon, the mechanical energy of the moon is conserved. Also, since the potential energy of the moon goes as 1/r^2, the potential energy is minimal at r_{max} and maximal at r_{min}. Thus, the kinetic energy (and therefore the velocity) is maximal at r_{max} and minimal at r_{min}. With the v_{min} now known (as explained in the previous paragraph), the mass of the planet, M, can be calculated using the conservation of energy:

{m {v_{max}}^2}/2 + {G m M}/{r_{min}}^2 = {m {v_{min}}^2}/2 + {G m M}/{r_{max}}^2. Note again that m cancels. This eliminates choice (B).

Now that the mass of the planet is known, the period, \tau, can be calculated using Kepler's third law:

\tau^2 = {4 \pi^2 a^3}/{G M}. This eliminates choice (D).

Only choice (A) remains.
wallace
2014-10-22 08:46:09		 It's much better than the official solution!
Alternate Solution - Unverified
travis.nicholson
2006-10-23 01:40:58		A more appropriate solution would be:

2a = r_min + r_max, where a is the semimajor axis. This elimiates choice (E).

The angular momentum of the moon is conserved because there are no external torques. Therefore l_min = l_max -> m v_min r_min = m v_max r_max -> v_min = {v_max r_max}/r_min, where l is the magnitude of the angular momentum, and m is the mass of the moon. This eliminates choice (C).

Since there is no external work being done on or by the moon, the mechanical energy of the moon is conserved. Also, since the potential energy of the moon goes as 1/r^2, the potential energy is minimal at r_max and maximal at r_min. Thus, the kinetic energy (and therefore the velocity) is maximal at r_max and minimal at r_min. With the v_min now known (as explained in the previous paragraph), the mass of the planet, M, can be calculated using the conservation of energy:

{m {v_max}^2}/2 + {G m M}/{r_min}^2 = {m {v_min}^2}/2 + {G m M}/{r_max}^2. Note again that m cancels. This eliminates choice (B).

Now that the mass of the planet is known, the period, \tau, can be calculated using Kepler's third law:

\tau^2 = {4 \pi^2 a^3}/{G M}. This eliminates choice (D).

Only choice (A) remains.
travis.nicholson
2006-10-23 01:50:57		 Sorry ... for some reason when I attempted to fix the syntax in this solution the page would not allow me to submit the edited version. Therefore, I posted the solution again.
Alternate Solution - Unverified
travis.nicholson
2006-10-23 01:16:21		The above solution is incorrect because of the second paragraph. This paragraph states that \tau is proportional to the radius, which is variable whereas \tau is a constant. In fact, \tau is proportional to the semimajor axis. Furthermore, the constant of proportionality in this relationship contains the mass of the planet, so \tau alone cannot be calculated via the reasoning in the second paragraph.
Prufrock
2013-09-23 15:02:03		 Strongly agree, the reasoning given is circular.
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