Question

A satellite is in a circular orbit around an unknown planet. The satellite has a speed of 1.75 104 m/s, and the radius of the orbit is 5.00 106 m. A second satellite also has a circular orbit around this same planet. The orbit of this second satellite has a radius of 8.75 106 m. What is the orbital speed of the second satellite?

Answer 1

Using Kepler's Third Law of Planetary Motion, the orbital speed of the second satellite was calculated to be approximately 14,800 m/s, based on the given orbital radii and the speed of the first satellite.

Step-by-step explanation:

To calculate the orbital speed of the second satellite, we'll use Kepler's Third Law of Planetary Motion, which states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. In the case of circular orbits, the semi-major axis is simply the orbit radius.

For two satellites orbiting the same planet, the ratio of their orbital speeds will be inversely proportional to the square root of the ratio of their orbital radii.

Let V₁ and R₁ be the speed and radius of the first satellite's orbit, and V₂ and R₂ the speed and radius of the second satellite's orbit. The formula relating these is:

V₂ = V₁ ∙ √(R₁/R₂)

Substituting the given values:

V₂ = 1.75 × 10⁴ m/s ∙ √(5.00 × 10⁶ m / 8.75 × 10⁶ m)

After calculation,

V₂ ≈ 1.48 × 10⁴ m/s

The orbital speed of the second satellite is approximately 14,800 m/s.

Answer 2

v = 1.32 10² m

Step-by-step explanation:

In this case we are going to use the universal gravitation equation and Newton's second law

F = G m M / r²

F = m a

In this case the acceleration is centripetal

a = v² / r

The force is given by the gravitational force

G m M / r² = m v² / r

G M/r = v²

Let's calculate the mass of the planet

M = v² r / G

M = (1.75 10⁴)² 5.00 10⁶ / 6.67 10⁻¹¹

M = 2.30 10²¹ kg

With this die we clear the equation to find the orbit of the second satellite

v = √ G M / r

v = √ (6.67 10⁻¹¹ 2.30 10²¹ / 8.75 10⁶)

v = 1.32 10² m