A satellite is in a circular orbit around an unknown planet. The satellite has a speed of 1.80 104 m/s, and the radius of the orbit is 5.20 106 m. A second satellite also has a circular orbit around this same planet. The orbit of this second satellite has a radius of 8.70 106 m. What is the orbital speed of the second satellite?
4π²R³ = T²GM and V²R = GM,
where R = radius of orbit in metres,
T = orbital period in seconds,
V = orbital speed in m/s,
G = gravitational constant ≈ 6.673×10−11 Nm²/kg²,
M = mass of Earth ≈ 5.98×1024 kg
To find the orbital speed of the second satellite, you can use Kepler's third law that states that the square of the orbital period of a satellite is directly proportional to the cube of its average distance from the planet.
Since the orbital period of both satellites is the same and constant, we can set up an equation using Kepler's third law:
(T₁ / T₂)² = (r₁ / r₂)³
Where T₁ and T₂ are the orbital periods of the first and second satellites, and r₁ and r₂ are the radii of their orbits, respectively.
Since the orbital periods are the same, we can omit them from the equation and solve for the ratio of radii:
(r₁ / r₂)³ = (5.20 × 10⁶ m / 8.70 × 10⁶ m)³
Simplifying the equation gives us:
(r₁ / r₂)³ = (676 / 373)³
Next, we can solve for the ratio of radii:
(r₁ / r₂) = 676 / 373 = 1.812
Now, we can use the ratio of radii to find the velocity of the second satellite. Since the velocity of the first satellite is given as 1.80 × 10⁴ m/s, we can use the following equation:
v₂ = v₁ * (r₁ / r₂)
Plugging in the values, we get:
v₂ = (1.80 × 10⁴ m/s) * (1.812)
Calculating the result gives us:
v₂ = 3.2616 × 10⁴ m/s
Therefore, the orbital speed of the second satellite is approximately 3.2616 × 10⁴ m/s.