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The masses and radii of the earth an moo...

The masses and radii of the earth an moon are `M_(1) and R_(1) and M_(2), R_(2)` respectively. Their centres are at a distacne d apart. Find the minimum speed with which the particle of mass m should be projected from a point mid-way between the two centres so as to escape to infinity.

Text Solution

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Potential energy of m when it is midway between `M_(1)` and `M_(2)`
`U = m(V_(1) + V_(2)) = m(-(GM_(1))/(d//2) + (-GM_(2))/(d//2))`
`= (-2Gm)/(d)[M_(1) + M_(2)]`
And as potential energy at infinity is zero, so work required to shift m from the given position to infinity,
`W = 0-U = 2Gm(M_(1) + M_(2))//d`
As this, work is provided by initial kientic energy.
`(1)/(2)mv^(2) = (2Gm(M_(1)+M_(2)))/(d)` or
`v = 2sqrt((G(M_(1)+M_(2)))/(d))`
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