A hydrogen atom, when vigorously perturbed, can emit light with a frequency of 6.16·1014 Hz. When the same light from hydrogen atoms in a distant galaxy is observed on earth, the frequency is 5.76·1014 Hz. Calculate the speed at which the galaxy is receding from the earth (in units of the speed of light, c).
To calculate the recession speed of the galaxy from the Earth, we can use the concept of the Doppler effect. The equation for the Doppler effect in terms of frequency is given as:
Δf/f = v/c
Where:
Δf = Change in frequency
f = Initial frequency
v = Relative velocity (recession speed) between the source and observer
c = Speed of light
In this case, the change in frequency (Δf) is given by:
Δf = observed frequency - initial frequency = 5.76·10^14 Hz - 6.16·10^14 Hz = -4.0·10^12 Hz
Plugging the values into the equation and solving for v:
-4.0·10^12 Hz / 6.16·10^14 Hz = v / c
Rearranging the equation to solve for v:
v = (-4.0·10^12 Hz / 6.16·10^14 Hz) * c
Now, we can substitute the value of the speed of light, c, which is approximately 3.0 x 10^8 meters per second:
v = (-4.0·10^12 Hz / 6.16·10^14 Hz) * (3.0 x 10^8 m/s)
Calculating the value of v:
v ≈ -1.95 x 10^6 m/s
Since the result is negative, it indicates recession, which means the galaxy is moving away from the Earth. The magnitude of the velocity can be obtained by taking the absolute value of v:
|v| ≈ 1.95 x 10^6 m/s
Finally, to express the velocity in units of the speed of light, c, we divide the magnitude of the velocity by the speed of light:
|v| / c = (1.95 x 10^6 m/s) / (3.0 x 10^8 m/s)
Simplifying the equation:
|v| / c ≈ 0.0065
Therefore, the galaxy is receding from the Earth at approximately 0.0065 times the speed of light.