Why is this specific definition of acceleration used in deriving a general equation for kinetic energy? $$\sum_{i=0}^n m_i v_i$$ as well as $$\frac{1}{2}\sum_{i=1}^n m_i v^2_i$$ Have been proved to be conserved by Newton & Leibniz, respectively. (http://en.wikipedia.org/wiki/Vis_viva) I don't understand how one would prove these conservations mathematically (is experiment necessary?)
Also does anyone know why $$\sum_{i=1}^n m_i v^2_i$$ was found to not be conserved? Some energy was being lost, obviously. Where did it go and how did people come to the realization that the loss was half of the energy?
Edit: Seeing as how the $\frac{1}{2}$ can be derived using Galileo's formulation for the study of the motion of uniformly acceleration objects, my new question is: Why is that definition of acceleration used when generalizing the kinetic energy equation to all accelerations (using integration)? Is there not a definition of acceleration with respect to position that is not based on the idea of uniform acceleration? 
The definition I'm talking about $$a_{avg} = \frac{1}{2}\frac{\Delta {v^2}}{\Delta x}$$ Where the $\frac{1}{2}$ comes from the kinematic equations
 A: I will give the more mathematical answer scince a very good description is already given.
Conservation of Momentum
Let's assume that we have a system of N particles. By definition these particles interact with each other and according to the Newton's 3rd Law, the forces of interaction are opposite.Therefore:
$$ \vec F_{ij}=- \vec F_{ji} $$
So, if we sum the internal Forces that are acted from every particle to every other particle of the system then we get:
$$ \sum_{i \neq j} \vec F_{ij}=\frac{1}{2}\sum_{i \neq j}( \vec F_{ij}+ \vec F_{ji})=\frac{1}{2}\sum_{i \neq j}( \vec F_{ij}- \vec F_{ij})=\vec 0 $$
From Newton's Second Law we know that:
$$ \sum \vec F_{total}=\sum \vec F_{external} + \sum \vec F_{internal}=\sum \vec F_{external}=\frac{d \sum \vec p}{dt}$$
Now if the system is isolated then $\sum \vec F_{external}=\vec 0$ so
$$\frac{d \sum \vec p}{dt}=\vec0 \Rightarrow \sum p =\sum_i m_iv_i = constant $$
As Benjamin pointed out the conservation of kinetic energy is not generally conserved. It's more complicated than that. A more fundamental theorem is the conservation of mechanical energy of a system, but only when all the forces that are acted upon it are conservative.
Potential Energy
When particles interact with conservative forces(interactions) meaning that:
$$ \oint \vec F \cdot d \vec r = 0 $$
then the field that describes the interaction can be dercribed by potential U. Because of this potential any particle/body with the right properties that enters this field gains potential energy.
We define potential energy as:
$$ V_{initial} - V_{final} = \int_{A}^{B} \vec F \cdot d \vec r $$
$$ dV = -dW $$
where A is the initial position and B is the final position.
As we can see, potential can only calculate differences between the initial and the final state. Therefore we can give any value to the initial state or the final. We chose to give the value 0 to the initial state and what we now get is:
$$ V_{final}= - \int_{A}^{B} \vec F \cdot d \vec r $$
Conservation of Mechanical Energy
Now lets assume that we have a system of particles and they interact with forces that we dont know if they are conservative or not. Then the work that all the internal forces produce is: 
$$ W_{total} = W_{interior} + W_{exterior}$$
$$ \int [ \sum \vec F_{in-conserv} + \sum \vec F_{in-non-conserv} + \sum \vec F_{ext-conserv} + \sum \vec F_{ext-non-conserv}]\cdot d \vec r $$
$$ \int \sum \vec F_{in-conserv} \cdot d \vec r + \int \sum \vec F_{in-non-conserv} \cdot d \vec r +\int \sum \vec F_{ext-conserv} \cdot d \vec r + \int \sum \vec F_{ext-non-cons} \cdot d \vec r $$ 
$$ W_{total} = W_{in-conserv} + W_{ext-conserv} + W_{in-non-conserv} + W_{ext-non-conserv} $$
$$ K_{int-f} - K_{int-in} + K_{ext-f} - K_{ext-in} = V_{int-in} - V_{int-f} + V_{ext-in} + V_{ext-f} +W_{int-non-cons}+W_{ext-non-cons} $$
If all the forces are conservative then:
$$ K_{int-f} + V_{int-f} + K_{ext-f} + V_{ext-f} = K_{int-in} + V_{int-in} + K_{ext-in} + V_{ext-in} $$
We define Mechanical energy as $E_M = K+V$ so:
$$ E_{M-ext-in} + E_{M-int-in} = E_{M-ext-f} + E_{M-int-f} $$
$$ E_{M-total-in} = E_{M-total-f} $$
So if all the forces that are acting on a system are conservative the total mechanical energy is conserved. Otherwise it's not.
Conservation of Kinetic Energy
Now, when the potential energy during a phenomenon does not change then we can talk about conservation of kinetic energy.
A: For the first summation (which corresponds to the Conservation of Total Linear Momentum) the condition to be met is that the "system" is expected to be isolated meaning that there should be no external forces acting from the rest of "Universe" onto the "system". This can be achieved from starting with the Newton's Second Law expressed in the form of: 
The total net force acting on a system is equal to the rate of change of the total linear momentum of the system. 
Where the total net force is expressed as a summation of internal and external forces. Since we assumed the system is isolated, the external forces would add up to zero. And, by Newton's Third Law you can show that the internal forces would also add up to zero. (because internal forces would add up to zero mutually according to Newton's Third Law).
Hence, as you can see, the left-hand side which is the total net force would be identically zero. This means that: The rate of change of Total Linear Momentum is zero which translates into Conservation of Total Linear Momentum.
The same method of approach also holds for the second summation (which corresponds to the Conservation of Total Kinetic Energy). However, this is more restrictive and tricky to prove because it requires more assumptions to be made about the "system;" it is usually not conserved for a "system" just because there is loss of kinetic energy into other forms of energies such as heat, sound and...This proof will be different than Conservation of Total Energy of the System which is more fundamental in Nature where ALL sorts of energies involved in the "system," are taken care of in the same manner as the first proof.
I hope this will give you a better picture of approaching the proofs. Thanks,
