Need help with the 2nd part of a frequency/ relative speed quesiton

[thundercats]

Homework Statement

A galaxy known to be traveling away from the Earth at 0.650 percent of the speed of light is observed on Earth to have a strong peak in its spectrum at 649.1 nm. However, one small portion of the galaxy (a "dwarf") has a similar observed peak with a wavelength that is 4.50 nm shorter. (This situation is not as strange as it may sound. Several galaxies can be observed to be in the process of collision or merger, such as the Whirlpool Galaxy. Several others, particularly radio galaxies, often exhibit high-speed jets or lobes.)
1) Calculate the wavelength of the light emitted by the bulk of the galaxy.
i understand this part of the question and got this answer 644.9 nm
What is the speed of the dwarf, relative to its "parent" galaxy?

Homework Equations

F'=F(1(+or-)U/C)

The Attempt at a Solution

i don't understand part 2 i have got part A i just don't know where and how to start

 

[anathema]

part B

To calculate the speed of the dwarf relative to its "parent" galaxy, you can use the Doppler effect equation:

f' = f(1 ± v/c)

Where f is the observed frequency, f' is the emitted frequency, v is the relative speed between the source and observer, and c is the speed of light.

In this case, we are given the observed wavelengths of the galaxy and the dwarf, and we know that the speed of light is constant. So, we can set up two equations:

f'galaxy = fg(1 + vgalaxy/c)
f'dwarf = fd(1 + vdwarf/c)

Since both the galaxy and the dwarf are emitting light at the same frequency (as indicated by the similar observed peaks in their spectra), we can set these two equations equal to each other:

fg(1 + vgalaxy/c) = fd(1 + vdwarf/c)

Now, we can solve for the relative speed between the galaxy and the dwarf:

vgalaxy/c = (fd/fg - 1)/(vdwarf/c)

Since we know the wavelengths of both the galaxy and the dwarf, we can plug those values in and solve for the relative speed. Remember to convert the wavelengths to frequencies using the equation c = fλ, where c is the speed of light, f is the frequency, and λ is the wavelength.

Once you have the relative speed, you can calculate the speed of the dwarf relative to its parent galaxy by simply subtracting the speed of the galaxy from the relative speed.

 

[thomate1]

part B

I would suggest starting by looking at the equation provided in the homework statement, F'=F(1(+or-)U/C). This equation relates the observed frequency (F') to the emitted frequency (F) and the relative speed (U) of the source to the observer. In this case, the observer is Earth and the source is the dwarf galaxy.

To solve for the relative speed of the dwarf, we need to rearrange the equation and solve for U. This can be done by dividing both sides of the equation by C and then taking the inverse sine of both sides. This will give us the equation U = C(sin(θ)-1), where θ is the angle between the direction of motion of the source and the observer.

Next, we need to find the angle θ. This can be done by using the information given in the problem. We know that the observed wavelength for the dwarf is 4.50 nm shorter than the observed wavelength for the bulk of the galaxy. Using the equation for wavelength, λ = c/f, where c is the speed of light and f is the frequency, we can set up the following equation:

λd = λb + Δλ

Where λd is the observed wavelength for the dwarf, λb is the observed wavelength for the bulk of the galaxy, and Δλ is the change in wavelength. We can rearrange this equation to solve for Δλ:

Δλ = λd - λb

Substituting in the values given in the problem, we get:

Δλ = 649.1 nm - 644.9 nm = 4.2 nm

Now, we can use the equation Δλ = λb (1 - cos(θ)) to solve for the angle θ. Rearranging the equation, we get:

θ = cos^-1(1 - Δλ/λb)

Substituting in the values, we get:

θ = cos^-1(1 - 4.2 nm/649.1 nm) = 0.389 degrees

Finally, we can plug this angle into the equation U = C(sin(θ)-1) to solve for the relative speed of the dwarf galaxy:

U = C(sin(0.389)-1) = 0.0087c

Therefore, the speed of the dwarf galaxy relative to its parent galaxy is 0.0087 times the speed of light