CK-12-Physics - Intermediate

22.3. Simultaneity http://www.ck12.org

wherev 1 is the velocity of an inertial frame(S′)measured in the proper frame(S), andv 2 is the velocity of an object
(or light wave) measured with respect to(S′). The relative velocityvris with respect to frame(S).

For example, if the above equation is applied to the Michelson-Morley experiment, we havev 1 = 30 kms ,the velocity
of the sun(S′)with respect to the earth(S)and the velocity of lightv 2 =cwith respect to the sun(S′). If we wish to
calculate the speed of lightvrrelative to the earth we have:

vr= 1 v+^1 +v 1 vv^22
c^2

→^30

kms+c

1 +^30

kmsc

c^2

=^30

kms+c

1 +^30

kms

c

=^30

kms+c

30 kms+c

c

=

( 30 km

s+c

1

)(

c

30 kms+c

)

=c.

This is, of course, the actual result of the experiment: the velocity of light iscregardless of the motion of the Earth.

Recall the example of the girl on the skateboard discussed earlier in the chapter. The girl has velocity 4. 0 ms relative
to the Earth, when she throws a ball with velocity 6. 0 ms relative to herself. The velocity of the ball relative to the
Earth,vrusing the relativistic equation for velocity addition is (when the ball is thrown in the positive direction),

v 1 = 4. 0

m

s

and v 2 = 6. 0

m

s

→

vr=

v 1 +v 2

1 +v^1 cv 22

=

4. 0 ms+ 6. 0 ms

1 +(

4. 0 ms)( 6. 0 ms)

c^2

=

10. 0 ms

1 +

(^24 ms^22

9 × 1016 ms 22

)

∼= 10. 0 m

s

Notice how small the factorv^1 cv 22 is compared to 1. In this non-relativistic case we havev^1 cv 22 = 2. 67 × 10 −^16. Unless
the quantityv^1 cv 22 is comparable to 1, the relativistic equation will reduce to the old Galilean equation for relative
velocity:

vr= 1 v+^1 +v 1 vv^22
c^2

≈v 1 +v 2.

In fact, all of the equations of special relativity will reduce to the classical ones whenvc. We typically regard
Newtonian physics as a low-speed case of the special theory of relativity. This in known as the principle of
correspondence. The principle of corresponded states that a general theory should always have, as a special case,
the theory which it supplants.

http://demonstrations.wolfram.com/EinsteinsFormulaForAddingVelocities/