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Question

A body is projected vertically upwards with a speed of GMR. The body will attain a height of
(M is mass and R is Radius of earth)

A
R2
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B
R
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C
54R
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D
3R2
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Solution

The correct option is B R
Since, there are no external or non-conservative forces acting on the system which is (body + earth), work done by them would be zero and hence mechanical energy of the system will be conserved.

Let m be the mass of the body.

Therefore, initial mechanical energy is given as

Ei=Ui+Ki=GMmR+mv22

Ei=GMmR+12m(GMR)

[ initial velocity is given as GMR]

Ei=GMm2R

And, final mechanical energy will be,

Ef=Uf+Kf

Ef=GMmr+0=GMmr

[ r is the maximum distance from the center of the Earth. At the highest point, velocity of the particle will be zero]

Conserving mechanical energy at earth surface and at the maximum height attained by the body,

PEi+KEi=PEf+KEf

GMm2R=GMmr

r=2RR+h=2R

h=R

Hence, option (b) is correct.
Key Concept - The gravitational potential energy for 2 point masses is given by U=Gm1m2/r

Why this question?
Note - A uniform spherical mass can be considered to be a point mass concentrated at its center to calculate gravitational potential energy, if the other mass is outside the sphere.

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