A satellite is put in an orbit just above the earths atmosphere with a velocity √(1 5) times the velocity - Physics - Gravitation

Question

A satellite is put in an orbit just above the earths atmosphere
with a velocity √(1 5) times the velocity for a circular
orbit at that height The initial velocity imported is horizontal
What would be the maximum distance of the satellite from the
surface of the earth when it is the orbit?
Ans in book is 2R

Kulshekhar Mehta · Accepted Answer

<img alt="" src=
"https://img-nm%20mnimgs%20com/img/study_content/content_ck_images/images/earth(2)%20png"
style="width: 253px; height: 251px;">
Given velocity of satellite(vi) when it is projected is
1 5voWhere vi is the initial velocity and vo is the velocity of
setelite in the circular orbit of radius (R+h)
Here we have to find h (maximum distance from the earth
surface)
According to energy conservation
EA=EQ⇒-GMmR+12mvi2=-GMmR+h+12mvo2⇒-GMmR+12m×1
5vo2=-GMmR+h+12mvo2⇒12m×1
5vo2-12mvo2=-GMmR+h+GMmR=GMm-R+R+hRR+h⇒12m×0
5vo2=GMmhRR+h⇒12×0 5vo2=GMhRR+h (1)And At point
QGMmR+h2=mvo2R+h⇒GMR+h=vo2 putting this valu in equation (1)
we have 12×0 5GMR+h=GMhRR+h 520=hR or, h=R4
Hence maximum distance from the earth surface when it is in orbit
is R/4