Question

The element sodium has a prominent spectral line at 588 nm. In a moving astronomical source, this line was measured at 589 nm. Is the object moving towards or away from the observer and how fast?

A) Moving away, 1 nm/s
B) Moving away, 2 nm/s
C) Moving towards, 1 nm/s
D) Moving towards, 2 nm/s

Answer 1

The astronomical object with a sodium spectral line shifted from 588 nm to 589 nm is moving away from the observer. The speed of the object is calculated using the Doppler effect formula, resulting in an approximate speed of 5100 m/s, which is closest to option B) Moving away, 2 nm/s.

Step-by-step explanation:

The element sodium has a prominent spectral line at 588 nm, but it was measured at 589 nm in a moving astronomical source. This phenomenon is known as the Doppler effect, which is the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. The shift to a longer wavelength indicates that the astronomical object is moving away from the observer.

To calculate the speed at which the object is moving, we use the formula for redshift (z):

z = (Δλ) / (λ0)
where Δλ is the change in wavelength (589 nm - 588 nm = 1 nm), and λ0 is the original wavelength (588 nm).

Thus, z = 1 nm / 588 nm = 1.70068 × 10−5. The speed (v) of the object can then be found using the formula v = z × c, where c is the speed of light (approximately 3 × 108 m/s).

Speed = 1.70068 × 10−5 × 3 × 108 m/s = 5102.04 m/s, which can be approximated to 5100 m/s when considering significant figures.

Therefore, the correct answer is: B) Moving away, 2 nm/s (since 1 nm/s is approximately 300 m/s, so 5100 m/s is closer to 2 nm/s).