If a star is receding (moving away from us) at a speed of 1.8% of the speed of light, at what wavelength would a spectral line with a laboratory wavelength of 422 nm be observed? Express your answer in nm to the nearest 0.1 nm.
To determine the observed wavelength of a star that is moving away from us, we need to apply the concept of the Doppler Effect, which states that the observed wavelength will be shifted depending on the relative motion between the source (the star) and the observer (us).
The formula for the observed wavelength is given by:
λ_observed = λ_lab * (1 + V/C)
Where:
λ_observed is the observed wavelength,
λ_lab is the laboratory wavelength,
V is the velocity of the star,
C is the speed of light.
In this case, the velocity of the star is given as 1.8% of the speed of light, and the laboratory wavelength is 422 nm.
First, we need to convert the velocity to a decimal fraction of the speed of light:
V = 1.8% * C
= (1.8/100) * C
= 0.018C
Now, we can substitute the values into the formula:
λ_observed = 422 nm * (1 + 0.018C / C)
= 422 nm * (1 + 0.018)
= 422 nm * 1.018
= 429.196 nm
Rounding to the nearest 0.1 nm, the observed wavelength of the spectral line would be approximately 429.2 nm.