Simple Nature - Light and Matter

coordinate axes. Even though ∆x, ∆y, and ∆z are different indifferently oriented coordinate systems, r is the same.g / Two balls roll down a cone andonto a plane.Kinetic energy example 55Kinetic energy equals (1/2)mv 2 , but what does that mean in threedimensions, where we have v x , v y , and v z ? If you were temptedto add the components and calculate K = (1/2)m(v x + v y + v z ) 2 ,figure g should convince you otherwise. Using that method, we’dhave to assign a kinetic energy of zero to ball number 1, sinceits negative v y would exactly cancel its positive v x , whereas ballnumber 2’s kinetic energy wouldn’t be zero. This would violaterotational invariance, since the balls would behave differently.The only possible way to generalize kinetic energy to three dimensions,without violating rotational invariance, is to use an expressionthat resembles the Pythagorean theorem,v =√v 2 x + v 2 y + v 2 z ,which results inK = 1 2 m (v 2 x + v 2 y + v 2 z).Since the velocity components are squared, the positive and negativesigns don’t matter, and the two balls in the example behavethe same way.192 Chapter 3 Conservation of Momentum