Escape velocity is the minimum speed that an object must...

Escape velocity is the minimum speed that an object must reach to escape the pull of a planet's gravity. Escape velocity $v$ is given by the equation $v=\sqrt{\frac{2 G m}{r}},$ where $m$ is the mass of the planet, $r$ is its radius, and $G$ is the universal gravitational constant, which has a value of $G=6.67 \times 10^{-11} \mathrm{m}^{3} / \mathrm{kg} \cdot \mathrm{s}^{2} .$ The mass of Earth is $5.97 \times 10^{24} \mathrm{kg},$ and its radius is $6.37 \times 10^{6} \mathrm{m} .$ Use this information to find the escape velocity for Earth in meters per second. Round to the nearest whole number. (Source: National Space Science Data Center)

Key Concepts

Escape Velocity

Escape velocity is the minimum speed an object needs to completely overcome the gravitational pull of a celestial body without further propulsion. It is derived by equating the object’s kinetic energy to the gravitational potential energy at the surface, emphasizing its dependence on the mass and radius of the celestial body.

Universal Gravitational Constant

The universal gravitational constant (G) is a fundamental physical constant that appears in Newton’s law of universal gravitation, governing the strength of the gravitational force between two masses. It serves as a scaling factor that ensures the gravitational force is accurately depicted for objects of various masses and distances.

Celestial Body Mass and Radius

The mass and radius of a celestial body are crucial parameters that determine its gravitational pull. In the context of escape velocity, a greater mass increases the gravitational binding energy while a larger radius indicates a lower gravitational field at the surface, both of which directly influence the minimum speed required for escape.

Energy Conservation in Gravitational Fields

The derivation of the escape velocity formula relies on the principle of energy conservation, where the kinetic energy imparted to an object must be sufficient to overcome the gravitational potential energy binding it to a celestial body. This balance of energies provides a clear method to calculate the escape velocity from a given gravitational system.

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