Two conservation laws but how are they compatible

[bburn]

Not a homework question, but I am trying to understand some stuff in Leibniz's 17th century "natural philosophy" which got me realizing that I don't have a clear grasp of my high school physics from 40 years ago.

Imagine a closed system consisting of 2 objects, and imagine a perfect elastic collision between them. The first object has a mass of 1 unit and is traveling at a velocity of 10 units (straight north). The second object weighs 2 units and is originally at rest. The first hits the second straight on and transfers all its motion to it. By the conservation of momentum, the second will travel at 5 units (same direction).

m1 * v1 = m2 * v2
1 * 10 = 2 * v2
v2 = 5

(I realize that velocity is a vector quantity, but if the direction is unchanged this can be ignored here, I think.)

Now, since the formula for Kinetic energy is one half mass times velocity squared, the kinetic energy of the system before the collision is (10 * 10) / 2 = 50 units, but the kinetic energy of the system after the collision is 2 * (5 * 5) / 2 = 25 units.

How is this consistent with the law of the conservation of energy? Has the energy been converted from kinetic energy to some other form? Or have I misunderstood something in my statement of the problem.

 

[Doc Al]

How is this consistent with the law of the conservation of energy?

It's not. What this tells you is that the first object cannot transfer all its momentum to the second in an elastic collision.

(You have to satisfy both conservation laws, not just conservation of momentum.)

 

[diazona]

The thing is, you can't just assume that the first object will stop and that all the momentum will be transferred to the second object. In your case, both of the objects will be moving after the collision, and it's the law of conservation of energy that can tell you exactly how fast they will be moving.

That's only really true for ideal objects, by the way - in a real collision, some of the energy does get converted into other forms, like sound.

 

[bburn]

thank you