The wavelength of light emitted from a distant galaxy by a hydrogen atom is observed to be 400 nm. If it is known that the laboratory wavelength of this light emitted by hydrogen is 380 nm, what is the velocity of the galaxy?
(a) .135c away from Earth
(b) .135c toward Earth
(c) .055 away from Earth
(d) .055 toward Earth
Redshifts, as is the case here, are away from the observer, and the wavelength shift is approximately
deltaL/L = v/c = 20/380 = .053
I used an approximate formula that is easier for me to remember than the correct relativistic Doppler formula, which you can find here:
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/reldop2.html
Go with (c)
To find the velocity of the distant galaxy, we can use the concept of redshift. Redshift occurs when light from an object moving away from us is stretched, causing its wavelength to increase.
The formula for redshift (z) is given by:
z = Δλ/λ,
where Δλ is the change in wavelength and λ is the original wavelength.
In this case, the observed wavelength (λ_observed) is 400 nm and the laboratory wavelength (λ_laboratory) is 380 nm.
So, the redshift (z) can be calculated as:
z = (λ_observed - λ_laboratory) / λ_laboratory
= (400 nm - 380 nm) / 380 nm
= 20 nm / 380 nm
= 0.0526
Redshift is related to the velocity (v) of the galaxy through the formula:
z = v / c,
where c is the speed of light.
Rearranging the formula, we have:
v = z * c
= 0.0526 * c
Now, we need to determine the sign of the velocity. If the redshift is positive (i.e., z > 0), it means the galaxy is moving away from us. Conversely, if the redshift is negative (i.e., z < 0), it means the galaxy is moving toward us.
Given that z = 0.0526, which is positive, the galaxy is moving away from us. Thus, the correct answer is (c) .055 away from Earth.