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Question

A projectile is projected with velocity kve in vertically upward direction from the ground into the space (ve is escape velocity and k<1). If air resistance is considered to be negligible, then the maximum height from the center of earth till which it can go will be (R= radius of earth).

A
Rk2+1
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B
Rk21
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C
R1k2
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D
Rk+1
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Solution

The correct option is C R1k2
Potential energy at surface Ui=GMmR
Potential energy at height h from the center is final Potential energy Uf=GMmh
Where M and m are mass of the earth and the projectile respectively.
Difference in potential energy is UfUi=GMmR(1R/h)
this difference in potential energy is nothing but the kinetic energy supplied at the time of projection

We know that escape velocity is 22GM/R so projection velocity v=k×22GM/R so the kinetic energy K.E.=mv2/2=mk2GM/R
equating the K.E. and the change in potential energy
we get h=R1k2
Option C is correct.

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