Speed as the fifth dimension In spacetime there are four general dimensions, three of space and one of time. Why is it that other dimensioned qualities seem to be rarely considered as part of spacetime? For example, why isn't speed part of spacetime, forming a five-dimension spacetime-speed manifold? Objects in spacetime change inertial reference frames much like they change position on any axis.
A moving object also appears distorted and reminds me of the distorted appearance of hypercubes when viewed from a lower number of dimensions.
I wouldn't think there's a rule against having a spacetime metric with composite dimensions added.
 A: We need to clarify what we mean by dimensions of a mechanical system. They refer, in the standard terminology, to the number of different degrees of freedom we need to describe the kinematics of a point particle (in this case). To this extend, given any reference frame $S$, an event in the space-time is identified by its position $(x,y,z)$ and the time $t$ it happens. The equations of motions for the dynamics of a point particle are (any relativistic form) of the Newton's laws, which define a second order degree Cauchy problem: as such, given the initial and boundary conditions, velocities can be calculated taking derivatives with respect to the path length $s$ as $v^{\mu}=dx^{\mu}/ds$ and therefore they are not independent variables.
In Hamiltonian and Lagrangian mechanics velocities can be treated as independent degrees of freedom unrelated to the positions, in principle. In fact the description of those systems is realised as a path on a $2n$-dimensional manifold, $n$ being the number of "position-like" degrees of freedom of the particle, which exactly fulfills the question you asked. Such manifold is the tangent (or cotangent) bundle $TM$ of the configuration space, which often undergoes the name of phase space. In general field theories, furthermore, one always defines the variation principle as an action whose variables are the fields $\phi(x)$ and its derivative $\dot{\phi}(x)$ treated as independent variables, giving rise to a twice dimensional dynamical manifold. Quantum field theory and string theory follow in some extends that flavour too.
A: I think Gennaro covered, this, I'll give a layman's explanation.

In spacetime there are four general dimensions, three of space and one
  of time. Why is it that other dimensioned qualities seem to be rarely
  considered as part of spacetime? For example, why isn't speed part of
  spacetime, forming a five-dimension spacetime-speed manifold? Objects
  in spacetime change inertial reference frames much like they change
  position on any axis.

Lets say you have a 3 body solar system, deep in empty space so effectively, the 3 bodies affect each other, but aren't affected by anything else.   Essentially, for our calculations, 3 objects in empty space.
With those 3 objects, you have a center of mass which doesn't change, but all 3 objects orbit around each other.  
Lets say you want to calculate the future locations of the 3 objects, in relation to each other.   This is an example of the 3-body problem or n-body problem.   Mathematically it's very complicated.  In fact, I think the 3 body problem remains unsolved, but lets ignore the mathematical complexity for now.
With 3 bodies, each gravitationally affecting the other 2, you need the following information to begin to work out the calculations.    Mass of each object, location of each object, Velocity and direction of each object (and direction in 3 dimensional space requires 3 variables).   So in a loose sense you're correct.   To get all the information about an object, 3 spacial dimensions and 1 time dimension doesn't tell you any information other than where and when.    So, in terms of information, velocity is an additional independent variable of information, which is different than a dimension, but direction of movement, together with velocity is actually 3 new variables)
So, for 1 body in a 3 body problem, you need to know location (X, Y, Z), time (T) and vector direction at that point in time (Vx, Vy, Vz) (the velocity in each direction combine to give you the total velocity, but to accurately calculate, you actually need 3 variables for velocity & direction).  
For an object in space, calculating the 3 body problem, you need 6 piece of information (X, Y, Z, Vx, Vy and Vx), at a specific time, and you need to know mass, to calculate gravitational effect on each other.   
These aren't spacial dimensions.   An object can only be in one place (3 dimensions) at one time (4th dimension), ignoring quantum behavior, ofcourse, where a particle can be in 2 places at once).
But in terms of solving the problem and figuring out where each object will be at a certain time in the future, you need more information than 3 spacial dimensions and time.  But at any point in time, an object can only be somewhere in the 4 spacial dimensions.   Velocity is irrelevant to location at a specific time.   It's necessary to calculate where it will be at a later time.
Forgive me if my words are clumsy, I don't think I'm explaining this very well.

A moving object also appears distorted and reminds me of the distorted
  appearance of hypercubes when viewed from a lower number of
  dimensions.

This is very different.   If you could actually see a 3 D slice of a rotating tesseract, you'd see a 3 dimensional object that didn't make any sense.  It would have straight edges and clean corners but as it turned, you'd see an object that kept changing shape.
A moving object doesn't behave that way at all.   At very high speed relative to you, objects will blue or red-shift, and you can see the far side of objects as light curves in around.   Objects can look stretched and warped at very high speed, but this is completely different than the weirdness that a theoretical 4 d object would look like.  A 4 D object would appear to be an object that can change it's shape without any apparent active force.  

I wouldn't think there's a rule against having a spacetime metric with
  composite dimensions added.

Mathematically, additional information can add dimensions to a numerical model (Kind of covered that above), but that's different than adding a dimension to space-time.   
Hope that helps.   I might try to clean this up later.
Edit - Space time is the model:
Space-time is a model, but it's not the model I was talking about.   In fact, the way this question is asked, I think Newtonian space or 4D space time isn't relelvant to the question.  3 dimensions of space and 1 of time, works in Newtonian Physics.
What I was going for, which maybe an example will help.   Something I remember from Grad School.   One of the more fun teachers I had, mentioned a technique that Lagrange (I think it was Lagrange), used to try to solve the 3 body problem.   Each of the 3 bodies has a location relative to the center of the 3 bodies, and 3 vector directions, and from there, you can calculate the gravitational force each of the bodies has on the other 2.    I think the way my teacher explained it was that each object has an X, Y and Z location, an X, Y and Z velocity and an X, Y and Z force acting upon it by the other 2 objects gravitational pull.
Lagrange, played with looking at the 3 body problem as a 27 dimensional space (Mathematical construct 9 bits of information per object) and the solution, I think he worked it down to a 13 dimensional object within that 27 dimensional space.    
All I meant was that using mathematical models, you can work mathematics in multiple dimensions, but that doesn't add a dimension to 3D space.   That's probably a confusing way to look at it and I could have said it simpler. 
For example, what I probably should have said: in space, Newtonian or Einsteinian, you need 3 bits of information for location and, while less obvious, you also need 3 bits of information for velocity.   That's a property of 3D space.   Each 3 bits of information correspond to the same 3 dimensions.   That's really all I meant to say.
A: Speed itself... is a type of "negative and/or anti length" and "negative and/or anti time"  dimension and/or probable "quantum" projection of "either/or". By formula, the amount of time used to transverse a distance is reduced by speed. As speed increases, the amount of time used decreases... and/or the length is, by first observer, shortened.
In order to achieve such, energy is expended. This energy, goes where? Into the one dimensional "place" where length itself resides in the fabric of space-time, effectively shortening it... and/or time. Of course, depending on "where" the observer is. The residual from the energy expended on the first observer, dilates time.
The "Speed Dimension' therefore, must be that which is in-between the Planck sized pieces of space-time. It is the "nothing" in-between, that is invoked, by energy, to undeniably, shrink the fabric of space-time.
A: A spacial dimension is a direction in which a body can move. This line of direction must lay 90° to other dimensions. The 3 dimensions commonly known to us are forward-back, up-down and front-back. A moving body in our universe used one or more of these directions when it moves. So far we have not detected any 4th spacial direction/dimension in our universe. There may be upto 11 dimensions (3 spacial, 1 time and 7 which we cannot feel or see) in our universe, but these 7 extra dimensions cannot be felt through our senses.
Speed cannot be the 5th dimension because speed does not introduce us to any different path of motion not included already in xyz space frame.
