How to "drop" a road (curve) to terrain (mesh)?

I have an serpentine road made of a curve, which goes in xy-plane. Below it i have made a terrain, it is a mesh, subdivided plane, having some mountains and hills. Is there a way to some how “drop” my curve (the road), to the terrain, so that it follows the terrain?

I have tried some softbody settings for the curve, but all i get is a jumping road. It is funny, but not quite what i needed.

How would you do this?

YOU COULD TRY RETOPO
you could select the base verticies of you road then retpo it to the ground and then select the rest of verticies and move them downward close to the ground

Salutatons