bei48482_FM

Figure 1.14 shows how pvaries with cfor both mand m. When cis small,

mand mare very nearly the same. (For 0.01c, the difference is only 0.005

percent; for 0.1c,it is 0.5 percent, still small). As approaches c, however, the

curve for mrises more and more steeply (for 0.9c, the difference is 229 percent).

If c, pm

, which is impossible. We conclude that no material object can

travel as fast as light.

But what if a spacecraft moving at  1 0.5crelative to the earth fires a projectile

at  2 0.5cin the same direction? We on earth might expect to observe the projec-

tile’s speed as  1  2 c. Actually, as discussed in Appendix I to this chapter, velocity

addition in relativity is not so simple a process, and we would find the projectile’s speed

to be only 0.8cin such a case.

Relativistic Second Law

In relativity Newton’s second law of motion is given by

F(mv) (1.19)

This is more complicated than the classical formula Fmabecause is a function

of . When c, is very nearly equal to 1, and Fis very nearly equal to mv, as it

should be.

d



dt

dp



dt

Relativistic

second law

Relativity 25

“Relativistic Mass”

W

e could alternatively regard the increase in an object’s momentum over the classical value

as being due to an increase in the object’s mass. Then we would call m 0 mthe rest

mass of the object and mm() from Eq. (1.17) its relativistic mass, its mass when moving rel-

ative to an observer, so that pmv. This is the view often taken in the past, at one time even

by Einstein. However, as Einstein later wrote, the idea of relativistic mass is “not good” because

“no clear definition can be given. It is better to introduce no other mass concept than the ‘rest

mass’ m.” In this book the term mass and the symbol mwill always refer to proper (or rest)

mass, which will be considered relativistically invariant.

4 mc

3 mc

2 mc

mc

0 0.2 0.4 0.6 0.8 1.0

Relativistic momentum

γmv

Classical momentum mv

Linear momentum

p

Velocity ratio v/c

Figure 1.14The momentum of an object moving at the velocity relative to an observer. The mass

mof the object is its value when it is at rest relative to the observer. The object's velocity can never

reach cbecause its momentum would then be infinite, which is impossible. The relativistic momen-

tum mis always correct; the classical momentum mis valid for velocities much smaller than c.

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