Question

Question #124ba

Answer 1

The correct answer is known only to three significant digits, `-9.50"km/s"`. Pick the option closest to that figure (I cannot see them during answering), and watch your sign!

Explanation:

First off, none of the answers is really correct. The wavelengths are not reported with enough accuracy to render all the listed digits significant! Here significant digits will be properly rendered and we will see where that leads.

We begin by noting what changes the wavelengths, namely it's the Doppler effect https://en.m.wikipedia.org/wiki/Doppler_effect).

When a police siren comes towards you, it is in effect running into its own waves and compressing them, making the waves shorter and higher in frequency. When the siren passes, the waves are stretched out to fill the expanding space behind the siren, so we hear the siren sound drop to a lower frequency with longer sound waves. The same is true with light, except our measurements have to be more precise because the speed of the moving star is small compared to the inherentvspeed of the light waves.

The amount the wavelength changes with light is given by the following formula, which accounts for relativity as well as the inherent changr in wavelength:

`(("Wavelength from moving object")/("Wavelength at rest"))^2=(1+(v/c))/(1-(v/c))`

(See: http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/reldop2.html.) Here `c` is the speed of light, 299,792 km/s, and `v` is the radial speed of the object which is positive when the object is going away from you (waves stretch out), negative when the object is coming towards you (waves are squeezed).

Now put the numbers in. The nanometer units in wavelengths cancel out so we omit them, and we assume both wavelengths are known to seven significant digits:

`(656.2642/656.2850)^2=(1+(v/c))/(1-(v/c))`

To seven significant figures:

`0.9999366=(1+(v/c))/(1-(v/c))`

Then cross multiply and combine like terms. Note what happens to the significant digits:

`0.9999366-0.9999366(v/c)=1+(v/c)`

`-1.9999366(v/c)=color(blue)(0.0000634)`

We have only three significant digits left on the right side after subtraction, since the numbers are not known to place values less than `10^(-7)`! So, contrary to all the answers given, we can render the result to only three significant digits, despite measuring the wavelengths to seven! Such is the world of experimental science.

Solve for `v/c` and then multiply by `c=299792"km/s"` to get:

`v=-9.50"km/s"`