You have been visiting a distant planet. Your measurements have determined that the planet's mass is twice that of earth but the free-fall acceleration at the surface is only one-fourth as large. To get back to earth, you need to escape the planet. What minimum speed does your rocket need?

Verified step by step guidance

1. Understand the problem: The escape velocity is the minimum speed required for an object to escape the gravitational pull of a planet without further propulsion. The formula for escape velocity is \( v_{e} = \sqrt{\frac{2GM}{R}} \), where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.
2. Relate the given information: The mass of the planet is \( M = 2M_{\text{Earth}} \), and the free-fall acceleration \( g = \frac{GM}{R^2} \) is given as \( g = \frac{1}{4}g_{\text{Earth}} \). Use this to find the radius \( R \) of the planet in terms of Earth's radius \( R_{\text{Earth}} \).
3. Set up the ratio for free-fall acceleration: \( \frac{GM}{R^2} = \frac{1}{4} \cdot \frac{GM_{\text{Earth}}}{R_{\text{Earth}}^2} \). Substitute \( M = 2M_{\text{Earth}} \) into the equation to solve for \( R \) in terms of \( R_{\text{Earth}} \).
4. Substitute \( M \) and \( R \) into the escape velocity formula: Replace \( M \) with \( 2M_{\text{Earth}} \) and \( R \) with the value derived in the previous step. The escape velocity formula becomes \( v_{e} = \sqrt{\frac{2G(2M_{\text{Earth}})}{R}} \).
5. Simplify the expression for \( v_{e} \): Use the relationship between \( R \) and \( R_{\text{Earth}} \) to express \( v_{e} \) in terms of Earth's escape velocity \( v_{e, \text{Earth}} = \sqrt{\frac{2GM_{\text{Earth}}}{R_{\text{Earth}}}} \). This will give you the escape velocity for the planet relative to Earth's escape velocity.

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. It is typically denoted by 'g' and varies depending on the mass of the planet and the distance from its center. On Earth, this value is approximately 9.81 m/s², but it can be different on other planets, as indicated in the question where the acceleration is one-fourth that of Earth's.

Escape Velocity

Escape velocity is the minimum speed an object must reach to break free from a planet's gravitational pull without any further propulsion. It is derived from the balance of kinetic energy and gravitational potential energy. The formula for escape velocity (v) is given by v = √(2GM/r), where G is the gravitational constant, M is the mass of the planet, and r is the radius of the planet. Understanding this concept is crucial for determining the speed needed for a rocket to leave the planet.

Mass and Radius Relationship

The relationship between mass and radius is essential in understanding gravitational forces and escape velocity. In the context of the question, the planet's mass is twice that of Earth, but the free-fall acceleration is one-fourth, suggesting a larger radius. This relationship affects the gravitational force experienced at the surface and is critical for calculating escape velocity, as both mass and radius are key variables in the escape velocity formula.