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

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First, determine the radius of the satellite's orbit. The radius is the sum of the Earth's radius and the altitude of the satellite. The Earth's average radius is approximately 6371 km. Therefore, the orbital radius \( r \) is \( 6371 \, \text{km} + 890 \, \text{km} \). Convert this total distance into meters for consistency in units.

Next, use the formula for the gravitational force acting as the centripetal force required for circular motion: \( F = \frac{G M m}{r^2} = \frac{m v^2}{r} \), where \( G \) is the gravitational constant \( 6.674 \times 10^{-11} \, \text{N} \, \text{m}^2/\text{kg}^2 \), \( M \) is the mass of the Earth \( 5.972 \times 10^{24} \, \text{kg} \), \( m \) is the mass of the satellite, \( v \) is the orbital speed, and \( r \) is the orbital radius.

Cancel the mass of the satellite \( m \) from both sides of the equation, as it appears in both the gravitational force and the centripetal force expressions. This simplifies the equation to \( \frac{G M}{r^2} = \frac{v^2}{r} \).

Rearrange the equation to solve for the orbital speed \( v \): \( v = \sqrt{\frac{G M}{r}} \).

Substitute the known values for \( G \), \( M \), and \( r \) into the equation to calculate the orbital speed \( v \). Ensure all units are consistent, particularly that \( r \) is in meters, to find the speed in meters per second.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Circular Orbit

A circular orbit is a path where a satellite moves around a celestial body in a circle. The gravitational force provides the necessary centripetal force to keep the satellite in orbit. Understanding circular orbits is crucial for calculating the orbital speed required to maintain a stable path around Earth.

Gravitational Force

Gravitational force is the attractive force between two masses, such as Earth and a satellite. It is given by Newton's law of universal gravitation, which states that the force is proportional to the product of the masses and inversely proportional to the square of the distance between them. This force is essential for determining the conditions for a satellite's orbit.

Orbital Speed

Orbital speed is the velocity a satellite must have to maintain its orbit around a planet. It depends on the mass of the planet and the radius of the orbit. For a satellite 890 km above Earth's surface, calculating the orbital speed involves using the formula derived from equating gravitational force to centripetal force, ensuring the satellite remains in a stable circular orbit.