Answer:
the lowest speed that can be measured is ± 1,828 10⁴ m/s
Explanation:
The expression for the relativistic doppler effect is
f₀ = fs √ (1-v / c) / (1 + v / c)
Where f₀ and fs are the observed and emitted frequency respectively, v is the velocity of the wave or particle
Let's use the relationship of the speed of the wave with its frequency and wavelength
c = λ f
f = c / λ
Let's replace
c / λ₀ = c / λs √ (1-v / c) / (1 + v / c)
λs/λ₀ = √ (1-v / c) / (1 + v / c)
We cleared the speed
(λs / λ₀)² = (1-v / c) / (1 + v / c)
(λs/λ₀)²2 = (c-v) / c + v)
(c + v) (λs / λ₀)² = c-v
v [1 + (λs / λ₀)²] = c [1 - (λs /λ₀)²]
v = c [1 - (λs /λ₀)²] / [1 + (λs / λ₀)²]
v = c (λ₀² - λs²) / (λ₀² +λs²)
Let's calculate the speeds for the two possible cases
When the wavelength increases
λs = 656.46 + 0.04 = 656.50 nm
λ₀ = 656.46 nm
(λ₀² - λs²) = 656.5² - 656.46² = 52.518
(λ₀² + λs²) = 656.5² + 656.46² = 861931.98
v = 3 10⁸ 52.518 / 861931.98
v = 1.828 10⁴ m / s
Case 2 the wavelength decreases
λ₀ = 656.46 - 0.04 = 656.42 nm
(λ₀² - λs²) = 656.42² - 656.46² = -52.5151
(λ₀² + λs²) = 656.42² + 656.46² = 861826.948
v = 3 10⁸ (- 52.5151) / 861826.948
v = - 1,828 10⁴ m / s
The negative sign indicates that the Source moves away from the observer
