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 r 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.

Explanation

Let the potential energy of the particle ` = E_p`
` therefore E_p = (-GM_1m)/((d//2)) + (-G M_2 m)/((d//2)) = - (2Gm (M_1 + M_2) )/(d)`
Let the escape velocity of the particle ` = v_e`
`therefore` Kinetic energy of particle `E_K = 1/2 mv_e^2`
Total energy ` = E_K + E_P`
or ` E = 1/2 mv_e^2 - (2Gm (M_1 + M_2))/(d)`
when the particle escapes to inifinity , its total energy E becomes zero .
` therefore 1/2 mv_e^2 - (2Gm(M_1 + M_2))/(d) = 0`
or `v_e^2 = (4G (M_1 + M_2))/(d) " or " v_e = 2 sqrt((G(M_1 + M_2))/(d))`