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Escape Velocity

• Definition: The minimum speed an object must reach to break free from the gravitational hold of a celestial body without further propulsion.

• Formula: [ v_e = \sqrt{\frac{2GM}{R}} ]

  • Where:

    • ( G ): universal gravitational constant (6.67 x 10^{-11} m^3 kg^{-1} s^{-2})

    • ( M ): mass of the celestial body (e.g., Earth, Moon)

    • ( R ): radius from the center of the celestial body to the point of escape

• Example Calculation for Earth:

  • Given:

    • Mass of Earth ((M) = 6 \times 10^{24} kg )

    • Radius of Earth ((R) = 6.4 \times 10^{6} m )

    • ( v_e = \sqrt{\frac{2 \times (6.67 \times 10^{-11}) \times (6 \times 10^{24})}{6.4 \times 10^{6}}} )

    • ( v_e \approx 7.9 \times 10^3 ms^{-1} \

Kinetic Energy Upon Falling

• Example: A 100 kg mass falling from infinity onto Earth.

  • Velocity at Impact: ( v = v_e = \sqrt{2gR} )

    • Where:

      • ( g = 9.8 m/s^2 )

      • ( R = 6400 \times 10^3 m )

    • Thus, ( v \approx 11.2 \times 10^3 m/s )

  • Kinetic Energy (KE): ( K.E. = \frac{1}{2}mv^2 ) ( K.E. = \frac{1}{2} \times 100 kg \times (11.2 \times 10^3 m/s)^2 \approx 6.27 \times 10^9 J )

Orbital Mechanics

• Satellite in Orbit:

  • Speed of a satellite close to Earth's surface is ( v = \sqrt{gR} )

    • Based on gravitational acceleration

• Escaping Saving Velocity for orbital satellites:

  • To overcome gravity while in orbit, additional speed imparted is determined:

  • ( v_{required} = v_e - v = 0.414 \times \sqrt{gR} )

• Gravitation Facts:

  • Kepler’s Laws govern the motion of planets and their relationship to the sun.

  • Gravitational potential energy ( U ) is defined as ( U = -\frac{GMm}{r} ) which is crucial for understanding celestial dynamics.

Objective Questions

• Example:

  • Escape velocity of a satellite in circular orbit is independent of its mass.

  • The mass and radius of a planet determine its gravitational influence on orbiting objects.

Assertive and Reasoning Questions

• Assertion: Kepler’s second law is based on the conservation of angular momentum.

• Reason: The areal velocity of orbital motion reflects the conservation principles in a gravitational system.

Numerical Examples & Applications

• Determining distances based on science principles helps understand escape velocity calculations.

• Understanding scaling in gravitational pull aids in visualizing mass relationships between celestial objects.

Summary Points

• Escape velocity relies solely on the mass and radius of the celestial body.

• The concept of gravitational potential and kinetic energy upon impacts enriches understanding of celestial dynamics in physical studies.