bei48482_FM

a moving object varies with its speed according to both classical and relativistic

mechanics.

The degree of accuracy required is what determines whether it is more appropri-

ate to use the classical or to use the relativistic formulas for kinetic energy. For in-

stance, when  107 m/s (0.033c), the formula^12 m^2 understates the true kinetic

energy by only 0.08 percent; when  3  107 m/s (0.1c), it understates the true

kinetic energy by 0.8 percent; but when 1.5  108 m/s (0.5c), the understate-

ment is a significant 19 percent; and when 0.999c,the understatement is a whop-

ping 4300 percent. Since 10^7 m/s is about 6310 mi/s,the nonrelativistic formula

^12 m^2 is entirely satisfactory for finding the kinetic energies of ordinary objects, and

it fails only at the extremely high speeds reached by elementary particles under cer-

tain circumstances.

1.9 ENERGY AND MOMENTUM

How they fit together in relativity

Total energy and momentum are conserved in an isolated system, and the rest energy

of a particle is invariant. Hence these quantities are in some sense more fundamental

than velocity or kinetic energy, which are neither. Let us look into how the total en-

ergy, rest energy, and momentum of a particle are related.

We begin with Eq. (1.23) for total energy,

Total energy E (1.23)

and square it to give

E^2 

From Eq. (1.17) for momentum,

Momentum p (1.17)

we find that

p^2 c^2 

Now we subtract p^2 c^2 from E^2 :

E^2 p^2 c^2  

(mc^2 )^2

m^2 c^4 (1^2 c^2 )



1 ^2 c^2

m^2 c^4 m^2 ^2 c^2



1 ^2 c^2

m^2 ^2 c^2



1 ^2 c^2

m



 1 ^2 c^2

m^2 c^4



1 ^2 c^2

mc^2



 1 ^2 c^2

30 Chapter One

bei48482_ch01.qxd 1/15/02 1:21 AM Page 30