Question

What will be the nature of the path for a particle projected with escape speed along the radius of the earth,away from it?

Answer 1

it should be a straight line because the particle is projected at escape vel., and by particle you mean it has negligible mass, so it'll not experience too much force and hence it'll move in a straight line.

Answer 2

I think it's not a straight line because even the force is small ,the mass is small and therefore you get the same acceleration