A starship is circling a distant planet of radius R. The astronauts find that the free-fall acceleration at their altitude is half the value at the planet's surface. How far above the surface are they orbiting? Your answer will be a multiple of R.

Verified step by step guidance

1. Step 1: Recall the formula for gravitational acceleration: \( g = \frac{GM}{r^2} \), where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( r \) is the distance from the center of the planet.
2. Step 2: Let \( g_s \) represent the gravitational acceleration at the planet's surface. At the surface, \( r = R \), so \( g_s = \frac{GM}{R^2} \).
3. Step 3: At the altitude of the starship, the gravitational acceleration is given as half the surface value, \( g_{orbit} = \frac{g_s}{2} \). Substitute \( g_s \) into this equation: \( \frac{GM}{r^2} = \frac{1}{2} \cdot \frac{GM}{R^2} \).
4. Step 4: Simplify the equation to find the relationship between \( r \) and \( R \): \( \frac{1}{r^2} = \frac{1}{2} \cdot \frac{1}{R^2} \). Rearrange to solve for \( r \): \( r = \sqrt{2} \cdot R \).
5. Step 5: The altitude above the surface is the distance from the center minus the planet's radius: \( h = r - R \). Substitute \( r = \sqrt{2} \cdot R \) to find \( h = (\sqrt{2} - 1) \cdot R \).

Key Concepts

Gravitational Acceleration

Gravitational acceleration is the acceleration experienced by an object due to the gravitational force exerted by a massive body, such as a planet. At the surface of a planet, this acceleration is denoted as 'g'. As one moves away from the surface, gravitational acceleration decreases according to the inverse square law, which states that the force of gravity is inversely proportional to the square of the distance from the center of the mass.

Orbital Mechanics

Orbital mechanics is the study of the motion of objects in space under the influence of gravitational forces. For a starship in orbit, the balance between gravitational pull and the ship's velocity creates a stable path around the planet. The altitude of the orbit affects the gravitational force experienced by the spacecraft, which is crucial for determining how far above the planet's surface the ship is located.

Height Above Surface

The height above the surface of a planet is the vertical distance from the planet's surface to the object in orbit. In this scenario, the astronauts experience a gravitational acceleration that is half of that at the surface, indicating they are at a specific altitude. This relationship can be expressed mathematically, allowing us to calculate the height in terms of the planet's radius, R, by applying the principles of gravitational acceleration and orbital dynamics.