Inelastic Collision and Energy How is is possible for momentum to be conserved and for KE to not be conserved? Momentum is related to velocity and velocity is related to KE, therefore if KE was lost, how can the system have the same momentum?
 A: That's easy. Think in a simple example that this happens. Imagine two particles of equal masses moving at $\vec v_1 = \vec v$ and $\vec v_2 = -\vec v$. Their momentum: $\vec p_1 = m\vec v_1$ and $\vec p_2 = m\vec v_2$.
The momentum of the system is therefore:
$$
\vec p = \vec p_1 + \vec p_2 = m\vec v_1 + m\vec v_2 = m\vec v - m\vec v = \vec 0
$$
The kinetic energy of the system is therefore:
$$
K = \frac{1}{2}mv_1^2 + \frac{1}{2}mv_2^2 \quad=\quad 
\frac{1}{2}m\vec v^2 + \frac{1}{2}m(-\vec v)^2 \quad=\quad 
mv^2
$$
Suppose now an inelastic collision happens, and all kinetic energy is gone. So, after collision, they are both with zero speed. And, after collision, the momentum is zero. Notice! There were conversation of momentum here, but all kinetic energy is gone.
The key reason here, is kinetic energy is quadratic on the velocities. The momentum is not. Which means, velocity $+\vec v$ or $-\vec v$ from the kinetic energy point of view has no difference. But there is a difference for momentum.
A: Kinetic energy is the work required to accelerate a mass from rest to a velocity (KE = 1/2 mv^2).  Momentum is a measure of the amount of movement a mass has at a velocity (p = mv).
Kinetic energy may be considered a process, and momentum may be considered the result of a process.  Momentum is conserved, but kinetic energy seems to come and go, as it changes from kinetic to potential when an object moves from speed to rest, or as it is expended to give an object momentum.  Therefore, though energy may change its form from kinetic to potential and back, energy is conserved.
Kinetic energy is a scalar, whereas momentum is a vector.  See this explanation of why kinetic energy is not considered to be a vector: http://www.quora.com/Why-isnt-kinetic-energy-a-vector-instead-of-a-scalar.  But although vectors are additive and seem naively easier to conserve, kinetic energy also is conserved, but in a different form.
