I know the totoal momentum and the total energy is conserved during the collision. And total energy is equal to the internal energy and the kinetic energy of the object. But I still can not see the internal energy from the formulea which only involve m and v. For me, it seems that if mv is conserved so should 1/2mv^2 be. It is just a little bit confusing that where does this 'internal energy' come from
"Internal energy" in the sense meant here is the energy that a system has even when that system as a whole is not moving. This can include a lot of things. In the usual case, two objects that collide and lose kinetic energy will gain thermal energy. They will be a little bit warmer after the collision. If you bend a metal rod repeatedly you can easily feel an example of this kind of heating. Depending on the scenario, the lost (or gained!!) energy in a collision can take the form of heat energy, chemical energy, electrical energy, light, sound, tension in springs, gravitational potential energy and many other things.
So what is momentum after all? What's the different between momentum and kinetic energy? Both of them only involve m and v, why do they turn out to be different?
They both involve m and v. However, v is different. Using the definition of kinetic energy, we say that KE = [itex]\frac{1}{2}[/itex]mv[itex]^{2}[/itex]. Here, v denotes the speed of the object, which is a scalar. Using the definition of momentum, we say that [itex]\vec{p}[/itex] = m[itex]\vec{v}[/itex]. Here, [itex]\vec{v}[/itex] is the velocity of the object, which is a vector. Thus, momentum is a vector and energy is a scalar. Also, energy and momentum are different because they mean different things. For example, if we wanted to put them in terms of standard SI units, we could say that momentum is measured in Newton seconds and energy is measured in Newton meters.
My book ( Chapter 10 ) says both are conserved: MV = m1v1 + m2v2 ( in explosion problems ) and Ktotal = 1/2 m1v1^2 + 1/2m2v2^2 but note ( for some reason ?? ) Ecm = 1/2 (m1 + m2) Vcm^2 is not equal to Ktotal I think your answer is somewhere in this mess ( book is Ohanian ). And I'm getting the book answers using these equations.
"Heat" energy is generated when something is broken or sticks in an irreversible way. Entropy is created. When entropy is created, the macroscopic forms of energy are not conserved. However, macroscopic momentum can still be conserved. A chemical bond, a cohesive bond, or some other bond is broken. In that case, the macroscopic kinetic energy is not conserved. The energy goes into random motions of the atoms. If you fire two cannon balls at each other, and the cannon balls stick together, bonds are formed. The two cannon balls may lose all their macroscopic kinetic energy. However, their temperature goes up. Thus, there is a lot of internal energy (sometime called heat energy) generated. The cannon balls may stop and fall. However, their total momentum hasn't changed. The combined momentum of the cannon balls was zero before and after the collision. The tricky part is knowing when heat energy is generated. In other words, one has to recognize when entropy is generated. I always look for something that breaks. If anything breaks, then I know thermal energy is generated. If nothing breaks, then thermal energy is not generated.
Not only can the macroscopic momentum be conserved, but it has to be. Because it's a linear quantity, it can't really "disappear" into internal degrees of freedom like energy can.