[UPSC IFoS 2022] Special Theory of Relativity (Q331, 8M)

Two spaceships are approaching each other as depicted below.

Spaceship 1 is moving with a velocity \vec{\text{u}}_1 = 0\cdot 7\text{c }\hat{\text{x}}, while spaceship 2 is moving with a velocity \vec{\text{u}}_2 = -\,0\cdot 8\text{c }\hat{\text{x}} as measured by the stationary observer on Earth. Here, c is the velocity of light in vacuum and \hat{\text{x}} is the unit vector along the x-axis.

(i) What, according to Galilean addition of velocity, is the speed of spaceship 2 with respect to spaceship 1 (in terms of c) ?

(ii) What, according to relativistic addition of velocities, is the speed of spaceship 2 with respect to spaceship 1 (in terms of c) ?

(iii) Spaceship 1 emits a light signal towards spaceship 2. The observer on spaceship 2 measures the frequency of the light signal to be 10^{16}\text{ Hz}. What was the original frequency of the light signal ?