Simple Nature - Light and Matter

g/Two balls roll down a cone and
onto a plane.

Pythagorean theorem equals

1

√

∆x^2 +∆y^2 +∆z^2

.

Rotating a line segment doesn’t change its length, so this expres-

sion comes out the same regardless of which way we orient our

coordinate axes. Even though∆x,∆y, and∆z are different in

differently oriented coordinate systems,ris the same.

Kinetic energy example 55

Kinetic energy equals (1/2)mv^2 , but what does that mean in three

dimensions, where we havevx,vy, andvz? If you were tempted

to add the components and calculateK= (1/2)m(vx+vy+vz)^2 ,

figure g should convince you otherwise. Using that method, we’d

have to assign a kinetic energy of zero to ball number 1, since

its negativevywould exactly cancel its positivevx, whereas ball

number 2’s kinetic energy wouldn’t be zero. This would violate

rotational invariance, since the balls would behave differently.

The only possible way to generalize kinetic energy to three di-

mensions, without violating rotational invariance, is to use an ex-

pression that resembles the Pythagorean theorem,

v=

√

vx^2 +vy^2 +vz^2 ,

which results in

K=

1

2

m

(

vx^2 +vy^2 +vz^2

)

.

Since the velocity components are squared, the positive and neg-

ative signs don’t matter, and the two balls in the example behave

the same way.

196 Chapter 3 Conservation of Momentum