Help Me Solve a Homework Problem Regarding Planet X's Satellites

[dearborne]

i was hoping somebody could give me a hand with a homework problem i have .

first here's the question .

-> Planet X has two satellites . A and B . Both orbits are circular . Satellite A's orbital radius is 4 times B's orbital radius . calculate the following ratios .

a) A's orbital period / B's orbital period
b) A's orbital speed / B's orbital speed


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now I'm kind of stumped because of the lack of information .
for a) i know that the ratio of orbital periods is the ratio of the T's . but I'm absolutly losing it trying to figure out how to find the T's .
i know that ((4pi^2)/G) (R^3/T^2) ) = M1 + M2
and R^3 = T^2 , also C = 2piR . but i can't for the life of me see how i could get the desired ratio with those formulas . does anybody know if I'm on the right track ? or should i be using some other formula that I'm somehoe missing ?

 

[BobG]

a) Kepler's third law says what. It explains the relationship between the radius and the orbital period. In other words, the relationship is explained by 2 key variables: T^2 and r^3. Everything else could be replaced by a single constant

b) Same reasoning. What's the relationship between speed and the radius. If the radius increases by a certain amount, does the speed increase or decrease? By how much?

Technically, you could let orbit B have a radius of '1' and orbit A have a radius of '4'. Then you could plug each into the appropriate equation, plug your answer into your ratio, and simplify. It's just easier to combine all of the constants into one called c, since the same constants will appear in both the numerator and the denominator and will cancel out.

Actually, the equation for each is:

$$\tau^2=\frac{4 \pi^2 r^3}{\mu}$$ with mu being the geocentric gravitational constant.

$$S = \frac{\sqrt{\mu}}{\sqrt{r}}$$

 

[thepatientmental]

Hello! I can definitely help you with this homework problem regarding Planet X's satellites. Let's start by reviewing the information given in the question. We know that Planet X has two satellites, A and B, both of which have circular orbits. The orbital radius of satellite A is 4 times that of satellite B. This means that the distance between Planet X and satellite A is 4 times greater than the distance between Planet X and satellite B.

To solve this problem, we need to use Kepler's third law, which states that the ratio of the squares of the orbital periods of two planets is equal to the ratio of the cubes of their semi-major axes. In simpler terms, this means that the ratio of the orbital periods (T) is equal to the square root of the ratio of the orbital radii (R).

So for part a), we can write the following equation:
(Ta/Tb) = √(Ra/Rb)

Since we know that the orbital radius of satellite A is 4 times that of satellite B, we can substitute in the values:
(Ta/Tb) = √(4/1) = 2

Therefore, the ratio of the orbital periods is 2.

For part b), we can use the formula for orbital speed, which is given by v = √(GM/R).
Since the satellites are orbiting the same planet, we can assume that the value for G and the mass of Planet X (M) are the same for both satellites. This means that the only variable that will affect the orbital speed is the orbital radius (R).

So, the ratio of orbital speeds can be written as:
(Va/Vb) = √(Ra/Rb)

Using the same values as before, we get:
(Va/Vb) = √(4/1) = 2

Therefore, the ratio of orbital speeds is also 2.

I hope this helps you solve the homework problem. If you have any further questions, please don't hesitate to ask for clarification. Best of luck!