A particle is at rest at a distance R(earth;s radius) above the earth's radius. The minimum speed with which it should be projected so that it should not return is 

this question can be solved by conservation of energy

 according to energy conservation ,total initial energy of system is always conserved...

initial potential energy=-GMm/2R                        (m is the mass object & M is mass of earth)

initial kinetic energy=mV_²/2

total energy intial=mV_²/2 - GMm/2R .....................1

finally, particle reaches to infinity &

potential energy is -GMm/infinity=0

final kinetic energy mV_1²/2

total energy final is m_V1²/2..............................2

equating  eq 1 and 2

   V= { GM/R + V_{1² }^1/2}

for minimum V ,V_{1 can be zero}

 so, V =(GM/R)^1/2