In 2000, NASA placed a satellite in orbit around an asteroid. Consider a spherical asteroid with a mass of 1.00×1016 kg and a radius of 8.50 km.

What is the speed of a satellite orbiting 4.70 km above the surface?
What is the escape speed from the asteroid?

1) set v^2/r=G*mass/r^2 where r is radius asteroid + altitude.

solve for v

2) 1/2 vescape^2=GMaste/radius

To find the speed of a satellite orbiting a celestial body, we can use the concept of gravitational force and centripetal force.

First, we need to calculate the gravitational force between the satellite and the asteroid. The formula for gravitational force is given by:

F = (G * M * m) / r^2

Where:
F is the gravitational force between the two objects,
G is the universal gravitational constant (6.6743 × 10^-11 Nm^2/kg^2),
M is the mass of the asteroid (1.00 × 10^16 kg),
m is the mass of the satellite (which we assume is negligible compared to the asteroid),
r is the distance between the satellite and the center of the asteroid (radius of the asteroid + altitude of the satellite).

Using the given values:

M = 1.00 × 10^16 kg
r = 8.50 km + 4.70 km = 13.20 km = 13,200 m

Plugging in these values into the formula, we can find the gravitational force between the satellite and the asteroid.

Next, to find the speed of the satellite in orbit, we can equate the gravitational force to the centripetal force:

F = m * v^2 / r

Where:
F is the gravitational force,
m is the mass of the satellite (which we assume is negligible compared to the asteroid),
v is the orbital speed of the satellite,
r is the distance between the satellite and the center of the asteroid (radius of the asteroid + altitude of the satellite).

By rearranging the formula, we can solve for the orbital speed:

v = √((F * r) / m)

Now, to find the escape speed from the asteroid, we can use the concept that the escape speed is the minimum speed required for a satellite to overcome the gravitational pull and escape from the gravitational field of the asteroid.

The escape speed is given by the formula:

v_escape = √(2 * G * M / r)

Where:
G is the universal gravitational constant (6.6743 × 10^-11 Nm^2/kg^2),
M is the mass of the asteroid (1.00 × 10^16 kg),
r is the radius of the asteroid.

Plugging in the given values into the formula, we can find the escape speed from the asteroid.

Now, let's calculate the values.

Gravitational Force:
F = (G * M * m) / r^2
F = (6.6743 × 10^-11 Nm^2/kg^2 * 1.00 × 10^16 kg) / (13,200 m)^2

Orbital Speed:
v = √((F * r) / m)

Escape Speed:
v_escape = √(2 * G * M / r)

By performing the calculations, we can find the answers to the given questions.