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Question

The masses and radii of the earth and moon areM1, R1 and M2, R2 respectively. Their centres are a distance d apart. The minimum speed with which a particle of mass m should be projected from a point midway between the two centres so as to escape to infinity is given by


A

2[G(M1+M2)md]12

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B

2[G(M1+M2)d]12

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C

2[G(M1M2)md]12

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D

2[G(M1M2)d]12

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Solution

The correct option is B

2[G(M1+M2)d]12


Let P be a particle of mass m situated midway between the centres of the earth and the moon as shown in the figure. The potential energy of particle P due to earth is
U1=GmM1r=GmM1d2=2GM1md
and that due to moon is U2=2GM2md

If the particle P is projected with a velocity v, its kinetic energy is K=12mv2. Therefore, the total initial energy of the particle is
Ei=U1+U2+K=2Gmd(M1+M2)+12mv2
If the particle is to escape to infinity, its total energy should be greater than or equal to zero. So, the minimum velocity required is given by
2Gmd(M1+M2)+12mv2=0
v=2[G(M1+M2)d]12
Hence, the correct choice is (b).


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