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Motion In Straight Line

Question
CBSEENPH11020254

A satellite is revolving in a circular orbit at a height ‘h’ from the earth’s surface (radius of earth R ; h<<R). The minimum increase in its orbital velocity required, so that the satellite could escape from the earth’s gravitational field, is close to: (Neglect the effect of atmosphere.)

  • square root of 2 gR end root
  • square root of gR
  • square root of gR divided by 2 end root
  • square root of gR space left parenthesis square root of 2 space minus 1 right parenthesis

Solution

D.

square root of gR space left parenthesis square root of 2 space minus 1 right parenthesis

Given, a satellite is revolving in a circular orbit at a height h from the Earth's surface having radius of earth R, i.e h <<R
Orbit velocity of a satellite
straight v equals space square root of fraction numerator GM over denominator straight R plus straight h end fraction end root space equals space square root of GM over straight R end root space left parenthesis as space straight h less than less than straight R right parenthesis
Velocity space required space to space escape
1 half space mv squared space equals space fraction numerator GMn over denominator straight R plus straight h end fraction

straight v apostrophe space equals space square root of fraction numerator 2 GM over denominator straight R plus straight h end fraction end root space equals space square root of fraction numerator 2 GM over denominator straight R end fraction end root space left parenthesis straight h less than less than straight R right parenthesis
therefore, the minimum increase in its orbital velocity required to escape from the Earth's Gravitational Field.
straight v apostrophe minus straight v space equals space square root of fraction numerator 2 GM over denominator straight R end fraction end root space minus square root of GM over straight R end root
space equals space square root of 2 gR end root minus square root of gR space equals space square root of gR space left parenthesis square root of 2 minus 1 right parenthesis