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Question

A projectile is fired vertically upwards from the surface of the earth with a velocity KVe, where Ve is the escape velocity and K<1. If R is the radius of the earth, the maximum height to which it will rise measured from the centre of the earth will be (neglect air resistance) :

A
1K2R
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B
R1K2
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C
R(1K2)
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D
R1+K2
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Solution

The correct option is B R1K2
Let the height attained by the projectile above the earth surface be h and radius of earth be R.
Thus distance of maximum height attained from the center of earth r=R+h ....(1)
Potential energy of the projectile at maximum height P.E=mgh1+hR
According to the law of conservation of energy : K.E=P.E
12mV2=mgh1+hR
12m(KVe)2=mg(rR)1+(rR)R (using 1)
Escape velocity at earth's surface Ve=2gR
12mK2(2gR)=mg(rR)1+(rR)R
K2R[1+(rR)R]=rR
K2r=rR
R=(1K2)r
r=R1K2

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