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. Figure 3.64 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 and (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.

a) $v_{32} = 2200 \textrm{ km/s}$, $v_{12} = -2300 \textrm{ km/s}$, $v_{42} = 5030 \textrm{ km/s}$, $v_{52} = 8900 \textrm{ km/s}$, $v_{55} = 0$

b) $v_{35} = -6700 \textrm{ km/s}$, $v_{15} = -11200 \textrm{ km/s}$, $v_{25} = -8900 \textrm{ km/s}$, $v_{45} = -3870 \textrm{ km/s}$, $v_{55} = 0$

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*3*after each of these subscripts here if you wanted, because they are all measured with respect to Galaxy 3. So that’s 2200 kilometers per second positive, and so that means from the perspective of somebody on Galaxy 2, Galaxy 3 appears to be moving to the right with this speed. Then we figure out what is the velocity of Galaxy 1 with respect to Galaxy 2, well it’s going to be the velocity of 1 with respect to 3 plus the velocity of 3 with respect to 2. We’ve written it this way so that the inner subscripts are the same and that’s why they cancel, leaving us with the outer subscripts, velocity 1 with respect to 2. So that’s negative 4500 kilometers per second plus 2200 kilometers per second that we’ve just calculated in the first part of this section a, and that gives negative 2300 kilometers per second is the velocity of Galaxy 1 according to an observer on Galaxy 2. Then Galaxy 4 with respect to 2 is velocity of 4 with respect to 3 plus 3 with respect to 2, the inner subscripts cancel giving us

*42*there for subscripts and that’s 2830 kilometers per second plus 2200 kilometers per second, giving us 5030 kilometers per second is the velocity of Galaxy 4 with respect to 2. Then velocity

*52*is velocity

*53*plus velocity

*32*, and this is 6700 plus 2200 which is 8900. Just to be complete, I put the velocity of Galaxy 2 with respect to itself but that’s a bit silly because velocity of something with respect to itself is always zero because you can’t move with respect to yourself. So then we do all of this again, but from the perspective of somebody on Galaxy 5. So the first step is to figure out what is the velocity of the Milky Way Galaxy, Galaxy 3 with respect to 5. It’s the negative of the velocity of 5 with respect to 3, so that’s the negative of this, so negative 6700 kilometers per second. So somebody in Galaxy 5 will perceive Galaxy 3 as moving this way with this speed. Then the velocity of Galaxy 1 with respect to 5 is the velocity of 1 with respect to 3 plus the velocity of 3 with respect to 5, so that’s negative 4500 minus 6700, and that gives negative 11200 kilometers per second.

*v25*equals

*v23*plus

*v35*, giving us negative 8900 kilometers per second.

*v45*is

*v43*plus

*v35*, and notice the pattern you’re seeing everywhere here that these inner subscripts are cancelling, leaving us with the outer subscripts, and that’s always the way it is with relative velocities. That makes negative 3870 kilometers per second and of course the velocity of the Galaxy 5 with respect to itself is zero.