Physics PE Question

[indianjatt]

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I had a question from TBR about PE.

One of the passages gave a formula for the PE for long range attractions. It was given as U = -GMEm/R.

One of the questions asked, what is the relationship between the PE of Spacecraft A if it is at an orbital radius of R to the PE of Spacecraft B if it as at orbital radius of 3R.
The choices were:
A 9 to 1
B 3 to 1
C 1 to 3
D 1 to 9
Answer: B

From my knowledge. the relationship between U and the Radius should be proportional b/c of the negative sign. I.e. if I have a higher Radius, my denominator and thus GMem/R ratio gets smaller, and a smaller negative number is larger than a large negative number.

B/C I thought U(PE) was proportional to the Radius. My assumption was that the higher R, the higher PE I would have and thus C would be the correct answer choice. Does anyone know why that is not the case here? Any explanation would be more than welcome!

 

[BigBlueBear]

Well...Spacecraft A will have a PE of let's say -A, and Spacecraft B will have a PE of (-1/3)A.

3:1 ratio.

Unless I'm missing something?

 

[Xiao]

Well...Spacecraft A will have a PE of let's say -A, and Spacecraft B will have a PE of (-1/3)A.

3:1 ratio.

Unless I'm missing something?

BigBlueBear's explanation is correct. Another way to look at this:

UA = potential energy of spacecraft A
UB = potential energy of spacecraft B
R = orbital radius

UA:UB = (-GMEm/R) : (-GMEm/(3R)) = 1/R : 1/(3R) = 1 : 1/3 = 3 : 1

The negative doesn't have anything to do with proportionality because it is cancelled out from both sides of the ratio. The key learning point here is the problem solving approach of applying the questions' given formula and setting up a proportionality equation.

EDIT: Gravitational potential energy is negative because the physics convention is that U=0 at R=infinite. If you do a calculus based derivation you will see why that makes sense from a mathematics perspective. Negatives don't have anything to do with proportionality for the MCAT. Directly proportional means (A ~ B), indirectly proportional means (A~1/B). Hope that helps.