Is it useful to geometrically represent conservation laws? In my physics class, we just studied collisions. We learned that momentum is conserved in collisions. I decided to examine a one dimensional collision in which two objects collide - object 1 and object 2.
We can put the masses of the colliding objects are put in to a vector
$\vec{m} =  \langle m_1,m_2\rangle $. We can also put the velocities of the colliding objects in to a vector $\vec{v} =  \langle v_1,v_2\rangle $, where the directions of $v_1$ and $v_2$ are given by their signs. Then, the momentum of the system is given by: $p = \vec{m}\cdot \vec{v}$, where the direction of the momentum is given by the sign of $p$.
Now we can express the conservation of momentum in this way:
$$
p = \vec{m}\cdot \vec{v}= \|\vec{m}\|\|\vec{v}\|\cos(\theta) = \text{constant}
$$
where $\theta$ is the angle between $\vec{v}$ and $\vec{m}$.
So, if the masses are constant, then if $\|\vec{v}\|\cos(\theta) $ is constant, momentum is conserved. 
If the mass and velocity vectors are based at the origin and the mass vector stays constant, then momentum is conserved as long as the projection of the velocity vector on to the mass vector remains constant. This defines a line along which the tip of the velocity vector can slide while conserving momentum.
Through this process the conservation of momentum has been geometrically represented, by a line in this case. With a bit of work, I think this could be done for other conservation laws and with more interacting objects. For instance, another curve could be drawn that corresponds to the conservation of kinetic energy. My question is: Is representing conservation laws in this way useful? That is, does it make understanding any situations easier or allow one to solve certain problems in a better way?
Also, if this technique is useful for anything, what sorts of topics could I research in order to learn more about it?
Edit:
Kyle Kanos made some good points below. I should clarify how this process results in the geometric expression of a conservation law. The ordered pair of masses $(m_1,m_2)$ and the ordered pair of velocities $(v_1,v_2)$ can be plotted in some two dimensional space as directed line segments based at the origin (which I was referring to as vectors above, but I will avoid doing so from now on). For the ordered pair of masses, the directed line segment would go from the point $(0,0)$ to the point $(m_1,m_2)$. Similarly, for the ordered pair of velocities, the directed line segment would go from the point $(0,0)$ to the point $(v_1,v_2)$. So, we have two directed line segments which represent the system in a certain configuration. I would call this a "geometric representation" of the system. 
Now, as shown above, as long as the projection of the velocity related directed line segment on to the mass related directed line segment remains constant, then the momentum of the system remains constant. So there is a set consisting of an infinite number of velocity related directed line segments for which the momentum of the system is the same. All of the terminal endpoints of these directed line segments fall upon a line. As long as the velocity related directed line segment has its terminal endpoint along this line, momentum is constant. So both before and after the collision, the velocity related directed line segment's terminal endpoint must be somewhere along this line. The conservation law restricts the terminal endpoint of the velocity related line segment to fall upon a line. This is a geometric expression of a conservation law.
 A: Your 1D "geometric" approach obviously fails in higher dimensions, as momentum is a vector quantity whereas you've described it as a scalar. The vectors you describe aren't spatial vectors of course, and so for higher dimensions (and more particles) they would seem to complicate things too much. Anyways, regarding the second part of your question, there is a very profound and interesting description of conservation laws that I think is geometric in the sense that you mean. It's called Noether's Theorem, and it relates the physical symmetries of a problem (rotational symmetry, translational symmetry, etc...) to physically conserved quantities. You'll no doubt learn about it when you study Lagrangian mechanics, but it might be worth your while to begin reading about it now. 
I'll sketch how it goes so you can get an idea of what it says. Given a certain quantity called a Lagrangian, defined as $L=T-U$, where $T$ is the kinetic energy of all the particles in your system, and $U$ is the potential energy of all the particles in your system, you can describe the conservation laws in terms of symmetries of this Lagrangian. For example, if the Lagrangian is independent of any spatial coordinates, so that $U=const.$, then linear momentum is conserved. This is called translational invariance, because you can move the particle around without affecting the Lagrangian. Another common example is the conservation of angular momentum. If the Lagrangian is unaffected by rotating the physical system by some angle, i.e. if the potential energy is only dependent on the radial distance $r$, then the total angular momentum of the system is conserved. 
There are much more exciting examples like conservation of energy (time translational invariance), and in particle physics, conservation of charge (phase invariance) which can be derived from these considerations. Wikipedia isn't a bad place to start if you want to read about it. Noether's Theorem
