Question

Please answer this Question :
Class 11 Gravitation .Physics .

Answer 1

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Answer : 1) Energy of the satellite in its new orbit is given as
$\frac{1}{2} m {v}^{2}$
Angular momentum of Satellite in new orbit :
2/√3 GMm/v

2)Orbit is Hyperbola with eccentricity :√(7/3)


3) Speed at closest distance to the Earth is
$\frac{v}{2} ( \sqrt{3} + \sqrt{7} )$


Now,
Here,
Effective mass is same as mass of satellite as Here it is assumed that Mass of Earth>>>Mass of Satellite .


Concepts involved :

1) Angular Momentum Conservation .

2) Total Energy Conservation .

3) If eccentricity of a curve is greater than 1, then that curve is Hyperbola.



Steps involved:


Use conservation of angular momentum to
find relationship between r(p) and r(a).



1) Initially, When Satellite was revolving in an Elliptical Orbit , we will use Energy conservation at two points namely,
Perigee and Apogee to find relationship between r(p) or r(a) and G,M and v.


2) Initial Energy of Satellite before launching of rocket is given by

PE at Apogee + KE at Apogee



Final Energy (after launching of rocket):

E(initial) + Change in Kinetic Energy at apogee





4)Use values of Energy and angular momentum after launching of rocket as given in Question to get
eccentricity .

5) Use Energy conservation and Angular momentum conservation in New Orbit with velocities as closest distance and at apogee.

From, here solve for speed at Closest distance.



For Calculation see Pic


Hope, you understand my answer and it may helps you.