Question

Imagine that martians launch a rocket toward the Earth at a great speed. While the rocket is traveling toward us, it will appear _____ than it actually is.

A. Larger

B. Darker

C. More blue

D. More red

Answer 1

The rocket would appear more blue than it actually is.

Step-by-step explanation:

Assume that the rocket was initially at a distance of
`d` away from the Earth. Let
`c` denote the speed of light. It would take a time of
`(d / c)` before the first crest of the light (an electromagnetic wave) from the rocket arrive at the Earth.

If
`T` denote the period of this EM wave. The next crest of this wave would leave the rocket at a time of exactly
`T\!` after the previous crest.

Let
`v` denote the speed at which the rocket travels toward the Earth. In that time period of
`T`, the rocket would have moved towards the earth by
`v\, T`. Thus, the next crest of this wave would need to cover a distance of only
`(d - v\, T)` (as opposed to
`d`, the distance required for the first crest) before reaching the Earth. Thus, this subsequent crest would need to travel for a distance of only
`(d - v\, T) / c` before reaching the earth.

Let
`t = 0` refers to the time when the first crest of the light from the rocket leaves the rocket. The next crest would leave the rocket at time
`t = T`.

• The first crest would arrive at the earth at
  `t = (d / c)`.
• The next crest would arrive at the earth at
  `t = T + ((d - v\, T) / c)`.

Thus, it would appear to an observer on the earth that the period of the light from the rocket was:

`\begin{aligned}& \left(T + (d - v\, T)/(c)\right) - \left((d)/(c)\right) \\ =\; & T - (v\, T)/(c)\end{aligned}`.

In other words, observers on the Earth would find the period of this wave shorter than it actually is at the rocket (
`T - (v\, T) / c` as opposed to
`T`.)

The frequency of a wave is inversely proportional to the period of that wave. Thus, observers on the Earth would find the frequency of the wave from the rocket higher than the frequency of this wave at the rocket.

EM waves of higher frequency (higher energy) appear more blue than ones of lower frequency. Therefore, this rocket would appear more blue to observers on the Earth than it actually is.