K.E. and momentum

Is K.E. proportional to momentum?
No

Why in collisions total momentum is conserved but not K.E.?
The conservation of momentum is implied by Newton's 1st law of motion and derived from the 3rd law of motion. This law becomes the law of nature at present. Another law of nature is the conservation of energy. The K.E. is not conserved but transferred to other energies like heat, work etc. K.E. becomes zero because velocity is zero.

Momentum mass x velocity is so defined as to describe objects in motion and relate it to define Force ( force is proportional to the rate of change of momentum)

K.E. 1/2mass x velocity square is so defined as to describe the energy of objects in motion, and related to Force by Work-done. Same for P.E.

Simply, Momentum --> Force --> Work --> Energy (K.E. or P.E.) or vise versa.

In fact in calculation of collisions, either momentum or energy conservation can be used.

2008-10-11 15:25:03 補充：
Your description of the objects are not clear so making this issue confusing. Your description above probably means the moving bodies are colliding with another inelastic body. Another situation is the moving inelastic bodies are colliding with each other only.

2008-10-11 15:27:23 補充：
IN the first case, the total momentum of the system includes the moving bodies + the inelastic body. So after collision the momentum of the moving bodies decreases as the momentum of the inelastic body increases, while the total momentum of the system remains unchanged.

2008-10-11 15:33:17 補充：
"Since inelastic collision, part of the energy is transfered to heat and sound, then the K.E. of of the moving bodies are decreased, therefore their total momentum decreases." is misleading. Only the momentum of the moving bodies decrease, not the total momentum.

2008-10-11 15:40:38 補充：
And the momentum decreases as a result of decrease in velocity, not K.E.

2008-10-12 22:08:46 補充：
cannot understand your question????
I repeat, momentum or K.E. are used separately to describe moving bodies, and there is no sense to relate both directly. Both are related as Momentum --> Force --> Work --> Energy (K.E. or P.E.) or vise versa. Please read in detail what I have presented above.

Q: Is K.E. proportional to momentum?

Kinetic energy (KE) and momentum increase with one another, i.e. an increase of momentum leads to an increase of KE and vice versa, but the increase is NOT proportional.

Q: Why in collisions total momentum is conserved but not K.E.? How does this work? Thank you.

It is because of the special physical property of momentum that in an isolated system, the quantity (momentum) is always conserved. The reason behind is that the impact force is related to rate of change of momentum with a proportional constant equal to unity. The independence of the proportional constant with mass and velocity of an object resulted in the conservation of momentum.

On the other hand, KE is only a form of energy. Energy is prone to change from one form to the other in any physical process. Collision is one of these processes that leads to the transfer of energy to other forms.