A planet of mass m moves in the inverse square central force field of the Sun of mass M . If the semi-major and semi-minor axes of the orbit are a and b , respectively, the total energy of the planet is:

          (a)    - ( G M m ) / ( a + b )

          (b)   - ( G M m ) [ ( 1 / a ) + ( 1 / b ) ]

          (c)    – [ ( G M m ) / a ] [ ( 1 / b ) - ( 1 / a ) ]

          (d)   - ( G M m ) / [ ( a – b ) / ( a + b )² ]

Solution:

Consider a planet is revolving around sun in elliptical orbit and the sun is situated at the centre of its orbit. Let the planet is moving with a linear velocity v₁ and v₂ when planet is at semi major axis ( a ) and semi minor axis ( b ).

Therefore,

Conservation of Momentum

L = m v₁ a = m v₂ b

By rearranging the terms we get,

v₂ = v₁ ( a / b )

 - - - - - - - - - - - - - - - - - - - - - - -   ( 1 )

Total energy of planet at semi major axis can be given as

E₁ = K.E.₁ + P.E.₁

E₁ = ( 1 / 2 ) m V₁² + [ - (  G M m / a ) ]

                                                                            - - - - - - - - - - - - - - - - - - - - - - - ( 2 )

Similarly,

Total energy of planet at semi minor axis can be given as

E₂ = K.E.₂ + P.E.₂

E₂ = ( 1 / 2 ) m V₂² + [ - (  G M m / b ) ]

According to law of conservation of energy,

E₁ = E₂ =

( 1 / 2 ) m V₁² + [ - (  G M m / a ) ] = ( 1 / 2 ) m V₂² + [ - ( G M m / b ) ]

( 1 / 2 ) m V₁² - ( G M m / a ) = ( 1 / 2 ) m V₂² - ( G M m / b )

Therefore,

( 1 / 2 ) m V₁² - ( 1 / 2 ) m V₂² = ( G M m / a ) - ( G M m / b )

From equation (1), above equation becomes

( 1 / 2 ) m V₁² - ( 1 / 2 ) m ( V₁ ( a / b ) )² = ( G M m / a ) - ( G M m / b )

( 1 / 2 ) m [ V₁² - ( V₁ ( a / b ) )² ] = G M m [ ( 1 / a ) - ( 1 / b ) ]

( 1 / 2 ) m V₁²  [ 1 - ( a / b )²  ] = G M m [ ( b – a ) / a b ]

( 1 / 2 ) m V₁²  [ ( b² - a² ) / b² ] = G M m [ ( b – a ) /  a b ]

As we known

( x² - y² ) = ( x – y ) ( x + y )

Therefore above equation becomes,

 ( 1 / 2 ) m V₁²   { [ ( b – a ) ( b + a ) ] / b² } = G M m [ ( b – a ) / a b ]

( 1 / 2 ) m V₁²  = G M m [ ( b – a ) / a b ] { b² / [ ( b – a ) ( b + a ) ] }

( 1 / 2 ) m V₁²  = G M m [ b /  a ] [  1 / ( b + a ) ]

But

K.E. ₁ = ( 1 / 2 ) m V ₁²

Putting this value in equation no (2) we get.

E₁ = E₂ = E = { G M m [ b / a] [ 1/ ( b + a ) ] }  - ( G M m / a )

E  = [ ( G M m ) / a ] { [ b / ( b + a ) ] – 1 }

E        = [ ( G M m ) / a ] { ( b – b – a ) / ( b + a ) }

E = [ ( G M m ) / a ] { ( a ) / ( b + a ) }

E = [ ( G M m ) / ( a + b ) ]

Answer : Total energy of planet is E = [ ( G M m ) / ( a + b ) ]