The escape speed from the surface of the Earth is 11.2 km / s . What would be the escape speed f

step 1 We know the escape velocity, but what is the escape velocity of another planet with twice the Ea

The escape speed from the surface of the Earth is 11.2 km /...

The escape speed from the surface of the Earth is $11.2 \mathrm{km} / \mathrm{s} .$ What would be the escape speed from another planet of the same density (mass per unit volume) as Earth but with a radius twice that of Earth?

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Key Concepts

Escape Velocity

Escape velocity is the minimum speed needed for an object to break free from the gravitational attraction of a celestial body without further propulsion. It is derived from the energy balance between kinetic energy and gravitational potential energy, leading to the formula v = sqrt(2GM/R), where G is the gravitational constant, M is the mass of the body, and R is its radius.

Mass-Volume Relationship

For spherical objects like planets, the mass is related to the volume, which in turn depends on the radius. Given a constant density, the mass of a planet is proportional to the cube of its radius (M ? R³). This relationship is critical when comparing how gravitational parameters change with the size of a planet of uniform composition.

Scaling Relations in Gravitational Systems

Scaling relations show how key quantities change when the parameters of a system are scaled. For instance, for planets of constant density, even though the mass increases as the cube of the radius, the escape velocity depends on the square root of the mass-to-radius ratio. Consequently, the escape velocity scales as the radius, which helps in understanding how changes in planetary size affect the gravitational pull.

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