Now imagine you’re an astronomer on Earth looking at the spectrum of an
object in space. In the spectrum you are looking for the so-called Lyman-�alpha line of hydrogen
which occurs at a wavelength of �� = 122 nm. In your telescope, you observe the Lyman-�alpha
line at 243 nm. What is the speed, in units of the speed of light, of the object?
(a) The object moves away from Earth at speed v = 0.60c
(b) The object moves towards Earth at speed v = 0.60c
(c) The object moves away from Earth at speed v = 0.99c
(d) The object moves towards Earth at speed v = 0.99c
To determine the speed of the object, we can use the formula for the Doppler shift. The Doppler shift formula for wavelengths is given by:
λ' = λ(1 + (v/c))
Where:
λ' = observed wavelength
λ = original wavelength
v = velocity of the object
c = speed of light
In this case, the observed wavelength (λ') is 243 nm, and the original wavelength (λ) is 122 nm.
Let's substitute the given values into the formula and solve for the velocity (v):
243 nm = 122 nm (1 + (v/c))
Divide both sides of the equation by 122 nm:
243/122 = 1 + (v/c)
2 = 1 + (v/c)
Subtract 1 from both sides:
2 - 1 = v/c
v/c = 1
Therefore, the speed of the object, in units of the speed of light, is equal to 1.
Comparing this result with the given options:
(a) v = 0.60c
(b) v = -0.60c
(c) v = 0.99c
(d) v = -0.99c
None of the options match the calculated value of 1. Therefore, none of the given options is the correct answer.
To determine the speed of the object, we need to use the equation known as the Doppler effect. The Doppler effect describes how the observed wavelength of light changes due to the relative motion between the source of light and the observer.
The equation for the Doppler effect is given by:
λ_observed = λ_emitted * (1 + (v/c))
Where:
λ_observed is the observed wavelength of the light
λ_emitted is the emitted wavelength of the light
v is the velocity of the object
c is the speed of light
In this case, the observed wavelength is 243 nm, and the emitted wavelength is 122 nm. We can substitute these values into the equation:
243 nm = 122 nm * (1 + (v/c))
Now, we can solve this equation for the speed of the object in terms of the speed of light (c).
243 nm / 122 nm = 1 + (v/c)
2 = 1 + (v/c)
To isolate v/c, subtract 1 from both sides:
2 - 1 = v/c
1 = v/c
So, the speed of the object, in units of the speed of light, is 1.
Now, let's compare this result to the answer choices provided:
(a) The object moves away from Earth at speed v = 0.60c.
This is not consistent with the derived value of 1, as it is less than 1.
(b) The object moves towards Earth at speed v = 0.60c.
This is not consistent with the derived value of 1, as it is less than 1.
(c) The object moves away from Earth at speed v = 0.99c.
This is also not consistent with the derived value of 1, as it is less than 1.
(d) The object moves towards Earth at speed v = 0.99c.
This is not consistent with the derived value of 1, as it is greater than 1.
None of the provided answer choices match the derived value of the speed, so none of the options are correct.