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The mass and radius of the earth are M1,...

The mass and radius of the earth are `M_1, R_1` and those for the moon are `M_2, R_2` respectively. The distance between their centers is d with that velocity should an object of mass m be thrown away from the mid-point of the line joining them so that it escapes to infinity ?

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`implies`  The distance between earth and moon is d. Since body is in between the line joining between earth and moon is Hence, its distance from earth or moon is `d//2` .
`implies` Total energy of body having stationary mass m at distance `d//2` due to earth. Potential energy `U_(1) = -(GM_(1) m)/(d//2) " "....(1)`
`implies` Total energy of body having stationary mass m at distance d/2 due to moon
Potential energy `U_(2) = - (GM_(2)m)/(d//2) " "...(2)`
`:.` Binding energy of body of mass m at the centre of earth,
` = (2Gm)/d [M_(1) +M_(2)]`
`implies` Suppose, a body escape to infinity at velocity `v_e.` hence escape energy,
`=1/2 mv_(e)^(2) " "...(3)`
`:. 1/2 mv_e^2 =(2Gm)/d [M_(1)+M_(2)]`
`:. v_(e)= sqrt((4G)/d(M_(1+M_2)))`
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