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. Identify the formula for gravitational acceleration at a distance r from the center of a planet, which is given by g(r) = \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. Set up the equation for the gravitational acceleration at the planet's surface, g_s = \frac{GM}{R^2}, where R is the radius of the planet.
3. Establish the relationship for the gravitational acceleration at the astronauts' altitude, which is half the surface value, g_a = \frac{1}{2}g_s = \frac{1}{2}\frac{GM}{R^2} = \frac{GM}{2R^2}.
4. Set the equation for gravitational acceleration at the astronauts' altitude equal to the general formula for gravitational acceleration at a distance r from the center, \frac{GM}{2R^2} = \frac{GM}{r^2}.
5. Solve for r by equating the denominators of the fractions and then solving for r in terms of R, which will give you r = \sqrt{2}R. Subtract R from r to find the altitude above the surface, h = r - R = (\sqrt{2} - 1)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 orbiting a planet, the balance between gravitational pull and the ship's velocity determines its orbit. The altitude of the orbit affects the gravitational acceleration experienced by the ship, which is crucial for understanding how far above the planet's surface the ship is located.

Height Above Surface

The height above the surface of a planet can be calculated using the relationship between gravitational acceleration at different altitudes. If the free-fall acceleration at the altitude of the starship is half that at the surface, we can use this information to derive the altitude in terms of the planet's radius. This involves applying the formula for gravitational acceleration and solving for the distance above the surface.