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Matej Velko
2014-Jan-22, 08:39 PM
Assume two bodies in complete isolation. The first body with mass {m}_{1 } is standing still which means its velocity {v}_{1 } is equal to 0. The second body with mass {m}_{2 } is approaching the first body with constant velocity {v}_{2 }. The collision is perfectly central and inelastic i.e. the two bodies are now one system with mass {m}_{1 }+{m}_{2 } and velocity v. The direction of \vec{v} is obviously the direction of \vec{{v}_{1 }}

The conservation of linear momentum certainly holds: {m}_{1 }{v}_{1 }=({m}_{1 }+{m}_{2 })v.

The conservation of energy states that no energy is lost so it should be the same before and after the collision: \frac{{m}_{1 }{{v}_{1 }}^{2 }}{2 }=\frac{({m}_{1 }+{m}_{2 }){v}^{2 }}{2 }.

If I express v from the last equation I get this: v=\sqrt{\frac{{m}_{1 }{{v}_{1 }}^{2 }}{{m}_{1 }+{m}_{2 } }}, however if I express v from equation of conservation of momentum I get this: v=\frac{{m}_{1 }{v}_{1 }}{{m}_{1 }+{m}_{2 } }.

How is it possible that I get different expressions for the same velocity when deriving it from the conservation of momentum and from the conservation of energy?

korjik
2014-Jan-22, 09:19 PM
Kinetic energy isnt conserved in an inelastic collision.

Matej Velko
2014-Jan-22, 09:25 PM
Kinetic energy isnt conserved in an inelastic collision.
To what other forms of energy is then the initial kinetic energy transformed into?

antoniseb
2014-Jan-22, 09:42 PM
Thermal. If two bodies hit each other and deform into a single object they will have heat equal to the energies of the two parts in the frame of reference of the resulting object.

EigenState
2014-Jan-22, 09:47 PM
Greetings,

If the two bodies formed a bond--think two atoms colliding to yield a molecule--some of the energy could certainly appear as vibrational excitation.

Best regards,
ES

tony873004
2014-Jan-23, 03:59 AM
Some energy is lost creating sound waves, which will ultimately end up as thermal energy. That's why dropping a marble is louder than dropping a superball. Superball's are about as close to a perfectly elastic collision as you can get, conserving most of the kinetic energy. They don't give up much of their kinetic energy to sound or thermal energy.

antoniseb
2014-Jan-23, 11:40 AM
Thermal. If two bodies hit each other and deform into a single object they will have heat equal to the energies of the two parts in the frame of reference of the resulting object.
... OK, I forgot to mention rotational energy if the two centers of mass were not perfectly aligned... and I was imagining the OP was about in a vacuum, but if not then tony873004's losses due to sound (vibrational damping from the surrounding medium) is also valid. On a macro scale, EigenState's molecular vibrations are heat.

Matej Velko
2014-Jan-23, 06:55 PM
... OK, I forgot to mention rotational energy if the two centers of mass were not perfectly aligned... and I was imagining the OP was about in a vacuum, but if not then tony873004's losses due to sound (vibrational damping from the surrounding medium) is also valid. On a macro scale, EigenState's molecular vibrations are heat.
I said the collision was perfectly central and yes, in vacuum but thanks for extra information.