The mass of Uranus is about 14.5 times that of the Earth, yet the escape velocity for Uranus is

step 1 As we all know that the acceleration due to gravity is given by G is equal to Gm by R square. Si

The mass of Uranus is about 14.5 times that of the Earth,...

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Key Concepts

Gravitational Potential Energy

Gravitational potential energy plays a crucial role in understanding escape velocity. The energy required to escape a body's gravitational field depends on how concentrated the mass is. A body with a lower density (mass spread over a larger volume) will provide a shallower gravitational potential well at its surface, allowing for a lower escape velocity compared to a denser body with a smaller radius.

Escape Velocity

Escape velocity is the minimum speed that an object must reach to break free from a celestial body's gravitational pull without further propulsion. It is determined by the formula v? = ?(2GM/R), where G is the gravitational constant, M is the mass of the body, and R is its radius. This formula shows that both the mass and the radius are key factors; while a higher mass increases the escape velocity, a larger radius decreases it.

Mass and Radius Scaling

When comparing celestial bodies, it is important to consider that an increase in mass does not automatically translate to a proportionate increase in escape velocity if that mass is spread over a larger radius. A larger radius means that the gravitational pull per unit distance at the surface is reduced, which can lead to a lower than expected escape velocity even if the mass is significantly higher.

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