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If a drop a ball on the earth I can describe itís motion using either F=ma or conservation of energy. If I hit two balls together on a pool table I need both F=Ma and conservations of energy to describe their motion.
My question is why do I need only a single equation in the first case and both equations in the second case. In the first case the equations seem redundant. In the second case they are both needed.
- Tony (age 39)
That's a really nice question.
For the falling ball there are no relevant 'hidden' energies. The only ones that count are the kinetic one and the gravitational potential. Since those are fixed by the speed and the height, you've got all the ingredients in either the kinematic (velocity, position) description or the energy decription. The force is just given by minus the spatial derivative of the potential energy, so it doesn't really whether you're given the force law or the energy law.
When the two balls collide there's another energy, the energy briefly stored in elastic deformations of the balls. These deformations are small and hard to keep track of. They arenít included in the standard kinematic description. However, they do hold a lot of the energy. If it's safe to assume that the energy almost all leaves those distortions and goes back into the large scale motion of the balls, use can use that fact together with momentum conservation, etc to figure out how the balls move after the collision even though you canít keep track of the forces during the collision.
(published on 10/22/2007)
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