For a satellite to be in a circular orbit 890 km above the surface of the earth, (a) what orbital speed must it be given?

Verified step by step guidance

1. Identify the radius of the orbit. The radius of the orbit (r) is the sum of the Earth's radius and the altitude of the satellite above the Earth. The Earth's radius is approximately 6371 km, so r = 6371 km + 890 km.
2. Use the formula for orbital speed, v = \sqrt{\frac{GM}{r}}, where G is the gravitational constant (6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2}), M is the mass of the Earth (approximately 5.972 \times 10^{24} \, \text{kg}), and r is the radius of the orbit in meters.
3. Convert the radius from kilometers to meters by multiplying by 1000, as the gravitational constant is in cubic meters per kilogram per second squared.
4. Substitute the values of G, M, and r into the formula to calculate the orbital speed.
5. The result will give you the orbital speed in meters per second, which is the speed the satellite must maintain to stay in a circular orbit at the specified altitude.

Key Concepts

Gravitational Force

The gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. For a satellite in orbit, this force provides the necessary centripetal force to keep it moving in a circular path. The strength of this force depends on the masses involved and the distance between their centers.

Orbital Speed

Orbital speed is the speed required for an object to maintain a stable orbit around a celestial body. It is determined by the balance between gravitational force and the object's inertia. For a circular orbit, the orbital speed can be calculated using the formula v = √(GM/r), where G is the gravitational constant, M is the mass of the Earth, and r is the distance from the center of the Earth to the satellite.

Altitude and Radius

Altitude refers to the height of an object above a reference point, typically the Earth's surface. In orbital mechanics, the radius is the distance from the center of the Earth to the satellite, which is the sum of the Earth's radius and the altitude of the satellite. Understanding the relationship between altitude and radius is crucial for calculating the orbital speed, as it directly affects the gravitational force acting on the satellite.