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14.12: Escape Velocity

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14.12: Escape Velocity

The escape velocity of an object is defined as the minimum initial velocity that it requires to escape the surface of another object to which it is gravitationally bound and never to return. For example, what would be the minimum velocity at which a satellite should be launched from the Earth's surface such that it just escapes the Earth's gravitational field?

To calculate the escape velocity, it is assumed that no energy is lost to any frictional forces. In practice, a satellite launched from the Earth's surface not only has to escape the Earth's gravitational field, but the Earth's atmosphere slows it down as well. Thus, the escape velocity calculated purely from gravitational energy considerations is smaller than the actual escape velocity.

It is also assumed that the satellite has zero velocity at infinity, where the Earth's gravitational force is zero. Since the escape velocity does not depend on the satellite's mass, it would be the same for any object, whether a satellite or a ball.

If an object is at a larger distance from the Earth's surface, for example, the Moon, it would need a smaller velocity to escape the Earth's gravitational field. This is unless the mass is directly heading towards the Earth and collides with it, giving rise to forces that are not just gravitational.

Alternatively, the escape velocity can be calculated by equating the total energy of a system, say the Earth and a satellite, to zero. The gravitational potential energy at large distances is conventionally assumed to be zero. Since the escape velocity is defined as the minimum velocity with which a satellite should be launched from the surface, its kinetic energy at infinity can also be assumed to be zero. Of course, if it is positive, it would go further away and never return. Thus, the velocity of interest is when the kinetic energy at infinity is zero; that is, the body just escapes the gravitational field.

Calculations reveal that the escape velocity from the Earth's surface, assuming no atmosphere, is about 11 km/s. In comparison, the escape velocity from the Sun's gravitational field is about 42 km/s at the Earth's distance. Spacecraft being launched from the Earth to escape the solar system would need to consider both factors.

Tags

Escape Velocity Object Initial Velocity Surface Gravitational Bound Never Return Satellite Launch Minimum Velocity Escape Earth's Gravitational Field Calculate Energy Frictional Forces Earth's Atmosphere Satellite Mass Distance Moon Collision Gravitational Forces Total Energy

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