phy1020.DVI

It is not obvious that this reduces to the classical expression until we expand into a Taylor series:

D



1 

v^2

c^2

1=2

D 1 C

1

2

v^2

c^2

C

3

8

v^4

c^4

C

5

16

v^6

c^6

C

35

128

v^8

c^8

C

63

256

v^10

c^10

C

231

1024

v^12

c^12

C (58.9)

Substituting this series expansion for into Eq. (58.8), we get

KD

1

2

mv^2 C

3

8

m

v^4

c^2

C

5

16

m

v^6

c^4

C

35

128

m

v^8

c^6

C

63

256

m

v^10

c^8

C

231

1024

m

v^12

c^10

C (58.10)

Unless the speedvis near the speed of lightc, all but the first term on the right will be very small and can be
neglected, leaving the classical equation.

Total Energy

If the only forms of energy present are the rest energyE 0 and the kinetic energyK, then the total energyE
will be the sum of these:

EDE 0 CKD

mc^2 : (58.11)

It is often useful to know the total energy of a particle in terms of its momentumprather than its velocityv.
It can be shown that the total energy is given in terms of momentum by

E^2 D.pc/^2 C.mc^2 /^2 : (58.12)

In the case where the total energy is much larger than the rest energy (E E 0 ), we may neglect the second
term on the right, and use

Epc: (58.13)