Question

The great astronomer Edwin Hubble discovered that all distant galaxies are receding from our Milky Way Galaxy with velocities proportional to their distances. It appears to an observer on the Earth that we are at the center of an expanding universe. The figure illustrates this for five galaxies lying along a straight line, with the Milky Way Galaxy at the center. Using the data from the figure, calculate the velocities:

(a) relative to galaxy 2

(b) relative to galaxy 5.

The results mean that observers on all galaxies will see themselves at the center of the expanding universe, and they would likely be aware of relative velocities, concluding that it is not possible to locate the center of expansion with the given information.

Short answer

a. The velocity of 1 relative to 2 is -2300 m/s. The velocity of 3 relatives to 2 is 2200 m/s. The velocity of 4 relative to 2 is 5030 m/s. The velocity of 5 relative to 2 is 8900 m/s.

b. The velocity of 1 relative to 5 is -11200 m/s. The velocity of 2 relatives to 5 is -8900 m/s. The velocity of 2 relatives to 5 is -8900 m/s. The velocity of 4 relative to 5 is -3780 m/s.

Step-by-step solution

Definition of velocity

Velocity is the rate of change in the position of an item in motion as seen from a specific frame of reference and measured by a specific time standard.

The velocity of galaxy 1 relative to galaxy 2 will be equal to the sum of the velocity of galaxy 1 relative to 3 and the velocity of the galaxy of galaxy 3 relative to 2.

The velocity of the galaxy 1 relative to 3 is V₁₃ = -4500 m/s.

The velocity of the galaxy 3 relative to 2 is V₃₂ = -2200m/s.

Hence V₂₃ = 2200 m/s.

Velocity of the galaxy relative to galaxy 2

Hence the velocity can be calculated by the following equation.

$$\begin{gathered} V_{12}=V_{13}+V_{32} \\ {V}_{12}=-4500+2200 \\ {V}_{12}=-2300\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right. \end{gathered}$$

The velocity of galaxy 1 relative to galaxy 2 is -2300 m/s

The velocity can be calculated by the following equation.

$$\begin{gathered} V_{32}=V_{23}+V_{32} \\ {V}_{32}=0+2200 \\ {V}_{32}=2200\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right. \end{gathered}$$

The velocity of galaxy 3 relative to galaxy 2 is 2200 m/s

The velocity can be calculated by the following equation.

$$\begin{gathered} V_{42}=V_{43}+V_{32} \\ {V}_{42}=2830+2200 \\ {V}_{42}=5030\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right. \end{gathered}$$

The velocity of galaxy 4 relative to galaxy 2 is 5030 m/s

The velocity can be calculated by the following equation.

$$\begin{gathered} V_{52}=V_{53}+V_{32} \\ {V}_{52}=6700+2200 \\ {V}_{52}=8900\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right. \end{gathered}$$

The velocity of galaxy 5, relative to galaxy 2 is 8900 m/s

Velocity of the galaxy relative to galaxy 5

Hence the velocity can be calculated by the following equation.

$$\begin{gathered} {\mathrm{V}}_{15}={\mathrm{V}}_{13}+{\mathrm{V}}_{35} \\ {\mathrm{V}}_{15}=-4500-6700 \\ {\mathrm{V}}_{15}=-11200\raisebox{1ex}{$\mathrm{m}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{s}$}\right. \end{gathered}$$

The velocity of galaxy 1 relative to galaxy 5 is -11200 m/s

The velocity can be calculated by the following equation.

$$\begin{gathered} V_{25}=V_{23}+V_{35} \\ {V}_{25}=-2200-6700 \\ {V}_{25}=-8900\raisebox{1ex}{$\mathrm{m}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{s}$}\right. \end{gathered}$$

The velocity of galaxy 2 relative to galaxy 5 is -8900 m/s

The velocity can be calculated by the following equation.

$$\begin{gathered} V_{35}=V_{33}+V_{35} \\ {V}_{35}=-6700 \\ {V}_{35}=-6700\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right. \end{gathered}$$

The velocity of 3 relative to 5 is -6700 m/s

The velocity can be calculated by the following equation.

$$\begin{gathered} V_{45}=V_{53}+V_{32} \\ {V}_{45}=2830-6700 \\ {V}_{45}=-3870\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right. \end{gathered}$$

The velocity of galaxy 4 relative to galaxy 5 is -3870 m/s