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?

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?