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I came across a question in the Princeton Review book for AP Physics 1 and then found the same question in the Collegeboard AP Physics course guide. Both have different answers. (See the questions below)

I understand that the distance can be calulcated using conservation of angular momentum, so that leaves just two answer choices. My question is about the reason that conservation of angular momentum can be used.

I know that the gravitational force on the moon is directed towards the planet, but what does that tell me about the conservation of momentum? Nothing... The direction of the gravitational force is NOT perpendicular to the path or velocity of the moon since it's in an elliptical orbit.

On the other hand, Newton's Third Law is connected to (or perhaps the basis of) the law of conservation of momentum. So shouldn't the answer be A for the Princeton question and D for the Collegeboard question? Am I wrong?

See the questions here:

Princeton Review Princeton Review Question

Princeton Review Answer

Collegeboard

Collegeboard Question Collegeboard Answer

Please help.

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    $\begingroup$ The two questions are exactly the same thing, and have the same answers (choice B). The Princeton review book has an error. $\endgroup$
    – knzhou
    Commented Mar 6, 2020 at 9:03
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    $\begingroup$ In general, prep books for high school exams are full of errors, sometimes up to a mistake every single page. They're not written by people who have any deep knowledge, and as you just saw, their practice questions are often made by taking real questions and (incorrectly) copying them. $\endgroup$
    – knzhou
    Commented Mar 6, 2020 at 9:03
  • $\begingroup$ If you're ever wondering who's right, <random prep book> or College Board, it's always the latter. $\endgroup$
    – knzhou
    Commented Mar 6, 2020 at 9:04
  • $\begingroup$ Thanks. But can I know why B is the correct answer? How does the direction of the gravitational force justify the conservation of momentum? $\endgroup$
    – MSayanvala
    Commented Mar 6, 2020 at 9:05
  • $\begingroup$ @MSayanvala The torque exerted by a force is given by $\tau = \vec{r}\times\vec{F}$. Because the force $\vec{F}$ is directed along the position vector $\vec{r}$ this cross-product vanishes and hence there is no torque. $\endgroup$
    – NDewolf
    Commented Mar 6, 2020 at 9:31

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If your frame of reference is placed on the center of mass of the planet, there is no torque exerted on the moon ($\mathbb{r}\parallel\mathbb{F}$), so angular momentum is conserved. At the points A and B the velocity of the moon is orthogonal to its position vector, so the magnitude of the angular momentum vector is $$|L|=m r_A v_A=mr_ B v_B.$$ At this point you just solve for $r_B$.
We used conservation of angular momentum and the fact that the force on the moon points towards the planet, so B is the right alternative.

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