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A particle is projected vertically upwar...

A particle is projected vertically upwards from the surface of earth (radius R) with a kinetic energy equal to half of the minimum value needed for it to escape. The height to which it rises above the surface of earth is

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The correct Answer is:
R
Applying energy conservation.
Total energy at A = Total energy at B
`1/2 m v^2 + (-"GMm"/R)=(-GMm)/(R+h)` ….(i)
Escape velocity = `V_e=sqrt((2GM)/R)`
Therefore, kinetic energy required for escape velocity
`1/2m v_e^2 =(GMm)/R`
The kinetic energy given is half of this value,
Therefore , `1/2m v^2 =(GMm)/(2R)`....(ii)
From Eqs. (i) and (ii) `(GMm)/(2R) - (GMm)/R = -(GMm)/(R+h) rArr - 1/(2R) =-1/(R+h) rArr R+h =2R rArr h=R`
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