In 2000, nasa placed a satellite in orbit around an asteroid. consider a spherical asteroid with a mass of 5.00�1015kg and a radius of 8.80km .a. what is the speed of a satellite orbiting 5.20km above the surface? b. what is the escape speed from the asteroid?

a).

b).

Explanation:

Using the conservation of energy and the kinetic energy of the satellite around the asteroid can model the motion in

a).

b).

a). What is the speed of a satellite orbiting 5.20km above the surface?

The speed of the satellite is .

b). What is the escape speed from the asteroid?

The escape speed from the asteroid is .

Explanation:

a). What is the speed of a satellite orbiting 5.20km above the surface?

The speed of the satellite can be found through the Universal law of gravity:

 (1)

Then, replacing Newton's second law in equation 1 it is gotten:

(2)

However, a is the centripetal acceleration since the satellite describes a circular motion around the asteroid:

 (3)

Replacing equation 3 in equation 2 it is gotten:

 (4)

Where v is the orbital speed, G is the gravitational constant, M is the mass of the asteroid, and r is the orbital radius.

Notice that the orbital radius will be given by the sum of the radius of the asteroid and the height of the satellite above the surface:

But it is necessary to express r in units of meters before it can be used in equation 4.

⇒

Hence, the speed of the satellite is

b). What is the escape speed from the asteroid?

The escape speed is defined as:

(5)

Where G is the gravitational constant, M is the asteroid radius and r is the orbital radius.

Hence, the escape speed from the asteroid is .

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