Question

Three particles are projected vertically upward from a point on the surface of the earth with velocities √(2gR/3),√(gR),√(4gR/3) respectively where R is the radius of the earth and g is the acceleration due to gravity on the surface of the earth. The maximum heights attained are respectively h1,h2,h3

A
h1:h2=3:4
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B
h2,h3=3:4
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C
h1,h3=1:4
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D
h2=R
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Solution

The correct option is A h1:h2=3:4Solution:-At maximum height, KE=0 so,KEi=12mv2, Ui=−GMmRKEf=0, Uf=−GMmgAs grating is a conservativeforce,KEi+Ui=KEf+Uf⇒12mv2−GmmR=−GMmh⇒h=−GMm12mv2−GMmR=GMGMR−V22So, h1=GMGMR−12(2GM3R)2/2=32Rh2=GMGMR−12(GMR)2/2=2Rh3=GMGMR−12(4GM3R)2/2−3Rh1:h2=3:4,h2:h3=2:3 and h1:h3=1:2

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