Answer to Question #98261 in Mechanics | Relativity for AbdulRehman

Question

By expanding equations (3.6) and (3.7) in a Taylor series, show that the
Newtonian and relativistic effects agree to first order in v
c =
vS−vR
c
. This
means that engineers who design radar devices do not in fact need to un-
derstand relativity. You can do this using the expansions from problem
(2-2). Note that “to first order” means that we ignore any terms propor-
tional to v
2
, v2
R, v2
S
or higher powers of v, vR, vS.

Answer 1

The relative Doppler effect:

$$\frac{\tau_R}{\tau_{S(rel)}}=\sqrt{\frac{c+v}{c-v}}.$$

Common Doppler effect:

$$\frac{\tau_R}{\tau_{S}}=\frac{c+v_S}{c-v_R}=\frac{c+v}{c}.$$

Here $v$ - velocity of the source relative to the resting receiver.

Expand them to first order:

$$T_1\bigg(\sqrt{\frac{c+v}{c-v}}\bigg)=1+\frac{v}{c},\\ T_1\bigg(\frac{c+v}{c}. \bigg)=1+\frac{v}{c}.$$