classical mechanics - What are the mathematical models for force, acceleration and velocity?

1. 
   

   In mechanics, the space can be described as a Riemann manifold. Forces, then, can be defined as vector fields of this manifold. Accelerations are linear functions of forces, so they are covector fields. But what about velocities and many other kinds of vectors?

   
   


2. 
   

   Of course velocities are not forces, so I don't think it is right to reuse vector fields of this manifold. But does this mean that this manifold has many different tangent spaces at each point?

   
   


3. 
   
   

   This sounds very strange to me. I think the problem is that math models have no physical units, maybe somehow we can create a many-sorted manifold to accommodate units?