Constant speed on 2d hills

Currently I have a straight terrain on which some units are walking. There is no physics involved. I want the terrain to have some sort of curvy hills and the units should climb and descend of. The path isn't always the same since there can be events like explosions that throws the units in the air.

What is the best way to achieve such a result. Should I add physics ? and if I do, how can I make sure that the units maintain the same speed when going uphill/downhill and how do I make them stick to the ground ? I also thought of using splines, but the problem here is when I want to add the explosion events.

I want to make something like Swords and Soldiers.

Hope the question is clear enough and I eagerly await for suggestion. I am new to Unity, but I have experience with other 2d game engines.