Question

A spaceship at rest with respect to the cosmic microwave background emits a beam of red light. A different spaceship, moving at a speed of $\mathbf{2}\mathbf{.}\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}$m/s towards the first ship, detects the light. Which of the following statements are true for observers on the second ship? (More than one statement may be correct.) (a) They observe that the light travels at $\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}$m/s. (b) The light is not red. (c) They observe that the light travels at $\mathbf{5}\mathbf{.}\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}$m/s. (d) They observe that the light travels at $\mathbf{2}\mathbf{.}\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}$m/s.

Short answer

Only (a) and (b) are true statements.

Step-by-step solution

Definition of Doppler effect

When a wave source travels relative to its observer, the Doppler Effect occurs, which causes a change in wave frequency. Christian Johann Doppler discovered it and described it as the process of starlight increasing or decreasing as a function of the star's relative motion.

Analyzing statement (a)

(a) This is true because the speed of light in vacuum is constant in all inertial reference frames and is equal to m/s. So, they will observe that light travels at $3\times {10}^8\mathrm{m}/\mathrm{s}$.

Analyzing statement (b)

(b) This is true because of Doppler effect. Since, the second ship is moving towards the first ship. The observers present there will witness that the light seems to be of blue color rather than red. They will observe this because the wavelength of light will decrease.

Analyzing statement (c)

(c) This is not true. The speed of light in vacuum is constant in all inertial reference frames and is equal to $3\times {10}^8\mathrm{m}/\mathrm{s}$. So, they will observe that light travels at role="math" localid="1657094370132" $3\times {10}^8\mathrm{m}/\mathrm{s}$.

Analyzing statement (d)

(d) This is not true. The speed of light in vacuum is constant in all inertial reference frames and is equal to $3\times {10}^8\mathrm{m}/\mathrm{s}$. So, they will observe that light travels at $3\times {10}^8\mathrm{m}/\mathrm{s}$.

Thus, out of all, the statements (a) and (b) are true.