TBR Physics Ch. 3, Discrete Q. 49

[vanillabear55]

Two identical spacecraft are orbiting a planet. Spacecraft A is at an orbital radius of R, while Spacecraft B is at 3R. What is the ratio of the gravitational potential energy of Spacecraft A to that of Spacecraft B?

A. 9 to 1
B. 3 to 1
C. 1 to 3
D. 1 to 9

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[exacto]

is the answer D?

 

[Odi]

Gravitational PE = U = -(G*M*m)/r

It asks for the ratio of A to B.

Therefore, you divide the Grav. PE formula of A by the Grav. PE formula of B.

A / B = [-(G*M*m)/ra] / [-(G*M*m)/rb]

Every variable is a constant except for radius so you can put a 1 for everything that's not changing.

A / B = [1/ra]/[1/rb]

Algebraically simplifies to rb/ra

They tell you rb = 3*ra

Further simplifies to [3*ra/ra]

ra's cancel and you have a 3:1 ratio.

You can answer it conceptually much faster...

R is in the denominator, therefore, as radius increases, the magnitude of U decreases. B is thus 3 times smaller in magnitude than A.

 

[popopopop]

^Thought grav. force equation was over r^2? 1^2 vs 3^2?

 

[Cawolf]

^Thought grav. force equation was over r^2? 1^2 vs 3^2?

It is, but potential energy is proportional to 1/r.

 

[popopopop]

Ahh, I see. New equation for me then.

 

[Cawolf]

Ahh, I see. New equation for me then.

Not really! The potential energy is the negative integral of the force. (Or the force is the negative derivative of the potential energy).

 

[popopopop]

Didn't take calc-based physics. I'll never learn what an integral is . *bows down*