Question

Scientists want to place a 4100.0 kg satellite in orbit around Mars. They plan to have the satellite orbit a distance equal to 1.5 times the radius of Mars above the surface of the planet. Here is some information that will help solve this problem:
mmars = 6.4191 x 1023 kg
rmars = 3.397 x 106 m
G = 6.67428 x 10-11 N-m2/kg2

What speed should the satellite have to be in a perfectly circular orbit?

Answer 1

Scientists want to place a 4100.0 kg satellite in orbit around Mars. They plan to have the satellite orbit a distance equal to 1.5 times the radius of Mars above the surface of the planet. Here is some information that will help solve this problem:

$$M _ {m a r s} = 6. 4 1 9 1 \cdot 1 0 ^ {2 3} k g; R _ {m a r s} = 3. 3 9 7 \cdot 1 0 ^ {6} m; G = 6. 6 7 4 2 8 \cdot 1 0 ^ {- 1 1} \frac {N \cdot m ^ {2}}{k g ^ {2}}$$

What speed should the satellite have to be in a perfectly circular orbit?

Solution.

The gravitational force between Mars and the satellite is given by:

$$F _ {g} = G \frac {M _ {m a r s} M _ {s a t}}{R ^ {2}},$$

where $R = R_{mars} + 1.5R_{mars} = 2.5R_{mars} = 8.49425 \cdot 10^{6}m$ is the distance between the center of Mars and the satellite; $M_{sat} = 4100kg$ is the mass of the satellite.

According to Newton's second law, this force produces a centripetal acceleration if the orbit of the satellite is perfectly circular:

$$F _ {g} = M _ {s a t} a,$$

where $a = \frac{v^2}{R}$ is the centripetal acceleration, $v$ is the speed of the satellite.

So we have an equation:

$$G \frac {M _ {m a r s} M _ {s a t}}{R ^ {2}} = M _ {s a t} \frac {v ^ {2}}{R}.$$

Find $v$ solving this equation:

$$v = \sqrt {\frac {G M _ {m a r s}}{R}} = 2, 2 4 6 \frac {k m}{s}.$$

Answer: $2,246\frac{km}{s}$