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A body is projected vertically upwards from the bottom of a crater of moon of depth R100 where R is the radius of moon with a velocity equal to the escape velocity on the surface of moon. Maximum height attained by the body from the surface of the moon is KR.The value of K is

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Solution


BC is the crater of moon. B denotes bottom of crater.

Depth of crater = R100

Radius of moon = R

Speed of particle at B=VB

Escape velocity on surface of moon =Ve

Ve=2GMR

At highest point VA=0

Mechanical energy is conserved in the process

Decrease in kinetic energy =12mV2B

Increase in PE=UAUB

VA=Potential at A=GM(R+h)

h = height from surface

Potential at B=
VB=GMR3[32R212r2]

=GMR3[1.5R20.5(RR100)2]

=GMR(1.01)

Increase in PE =GMmR+h+GMmR×(1.01)

Decrease in KE = increase in PE

12mV2B=GMm(R+h)+GMm×1.01R

GMR=GM[1R+h+1.01R]

or 1R=1R+h+1.01R

or h=99R

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