Question

Mass calculation question:
A satellite has an elliptical orbit around a an object. At its closest point, the satellite is 49km from the object and is traveling at 7.5 m/s. At it's farthest point, it's 98km from the object. What's the mass of the object?

Answer choices are:
5.05 x 10 ^16 kg
9.87 x 10 ^16 kg
4.25 x 10 ^16 kg
3.1 x 10 ^16 kg
2.04 x 10 ^16 kg

Since this isn't a circular orbit, I'm not sure how to determine the mass using the standard GM1M2/r^2 formula. If I calculate the mass at the closest approach, I get 4.16 x 10 ^16 kg...which I don't think is close enough for me to

Answer 1

be on the right track....any tips on how to calculate this?

Answer 2

Disregard...the equation to find M should be:
\[v1=\sqrt{\frac{2GM\frac{r2}{r1}}{r1+r2}}\]

Where v1 =7.5 m/s, r1 = 49000m, r2 = 98000m

Solving for M you get ~3.1 x 10^16

Answer 3

\[E=-k/2a =-k/r1 + (1/2) m v ^{2} \]
where E is the total energy, r1 and r2 are the pericenter and apocenter, respectively, and a is the semimajor axes of the ellipse (2a = r1+r2) (see Marion and Thornton, Ch 8). solving for k (k=mMG) we finally get:
\[M=(v ^{2}/2G)[(r1+r2)r1/(r1+r2-r1)]\]
with \[G=6.673\times10^{-11} Nm ^{2}Kg ^{-2}\]
we get \[M \approx3.1\times10^{16}Kg\]