A satellite is in a circular orbit around an unknown planet. The satellite has a speed of 1.55 104 m/s, and the radius of the orbit is 5.50 106 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?

You can do this with Kepler's law, or with mechanics.

With mechanics

FIrst satellite
gravity force=centripetal force.
GMm/5.5E6^2= mv^2/5.5E6
solve for GM

Next satellite, do the same thing, but you know GM, so solve for v

To find the orbital speed of the second satellite, we can use the principle of conservation of angular momentum.

The formula for angular momentum is given by:

L = mvr

Where L is the angular momentum, m is the mass of the satellite, v is the orbital speed, and r is the radius of the orbit.

Since both satellites are in circular orbits around the same planet, their masses and the gravitational force acting on them are the same. Therefore, we can equate the angular momentum of the first satellite to the angular momentum of the second satellite:

m1v1r1 = m2v2r2

Where m1, v1, r1 are the mass, orbital speed, and radius of the first satellite, and m2, v2, r2 are the mass, orbital speed, and radius of the second satellite.

We are given that:

v1 = 1.55 × 10^4 m/s
r1 = 5.50 × 10^6 m
r2 = 8.70 × 10^6 m

We need to find v2.

Substituting the values into the equation:

m1v1r1 = m2v2r2

We can cancel out the mass:

v1r1 = v2r2

Rearranging the equation to solve for v2:

v2 = (v1r1) / r2

Substituting the values:

v2 = (1.55 × 10^4 m/s * 5.50 × 10^6 m) / (8.70 × 10^6 m)

Calculating:

v2 ≈ 9.8 × 10^3 m/s

Therefore, the orbital speed of the second satellite is approximately 9.8 × 10^3 m/s.

To find the orbital speed of the second satellite, we can use the concept of conservation of angular momentum. The angular momentum of an orbiting object remains constant as long as there is no external torque acting on it.

The formula for angular momentum is:

L = mvr

where L is the angular momentum, m is the mass of the satellite, v is the orbital speed, and r is the radius of the orbit.

We can rearrange this equation to solve for v:

v = L / (mr)

Given that the first satellite has a speed of 1.55 x 10^4 m/s and a radius of 5.50 x 10^6 m, we can calculate its angular momentum and mass.

The angular momentum (L1) of the first satellite is given by:

L1 = m1v1r1

Similarly, for the second satellite with a radius of 8.70 x 10^6 m, we can calculate the angular momentum (L2) as:

L2 = m2v2r2

Since the angular momentum is conserved, we can equate the two expressions:

m1v1r1 = m2v2r2

Dividing both sides of the equation by (m2r2), we get:

v2 = (m1v1r1) / (m2r2)

Now we can substitute the given values into this equation to find the orbital speed of the second satellite:

v2 = (m1 * 1.55 x 10^4 m/s * 5.50 x 10^6 m) / (m2 * 8.70 x 10^6 m)

Unfortunately, the masses of the satellites are not provided in the question. Without the mass values, it is not possible to calculate the orbital speed of the second satellite.