Suppose we have a classical particle in box. The number of degrees of freedom is 6. The position of the particle and its momenta. Now if we want to calculate the entropy of the system as a function of the energy we only need to find a relation between all the possible states the particle can be in and its kinetic energy. The possible states in momentum space is proportional to the surface area of a sphere with radius p, where E=p^2/(2m). But what I don't understand is this: Why do the number of possible states in momentum space increase with an increase in the kinetic energy of the particle? Sure, the area of the sphere increses with an increase in radius. But why can't we just say that the number of possible states is proportional to the direction of the vector p from the origin? In this case the number of directions which would be the same as the number of possible states of p does not increase with an increase in energy. The number of possible directions is the same independent of the magnitude of p. So why do we say that the number of possible states increases when the particles momentum increases?