Question

Engineering Application It has been found that, on average, galaxies are moving away from Earth at a speed that is proportional to their distance from Earth. This discovery is known as Hubble’s law, named for its discoverer, astrophysicist Sir Edwin Hubble. He found that the recessional speed v of a galaxy a distance r from Earth is given by v = Hr, where H = 1.58  10-18 s-1 is called the Hubble constant. What are the expected recessional speeds of galaxies (a) 5.00  1022 m from Earth, and (b) 2.00  1025 m from Earth? (c) If the galaxies at each of these distances had traveled at their expected recessional speeds, how long ago would they have been at our location?

Answer 1

a)-b) We can find the expected recessional speeds of galaxies as follows:

$$v_1=Hr_1=1.58\cdot10^{-18}\ s^{-1}\cdot5.0\cdot10^{22}\ m=7.9\cdot10^4\ \dfrac{m}{s},$$ $$v_2=Hr_2=1.58\cdot10^{-18}\ s^{-1}\cdot2.0\cdot10^{25}\ m=3.16\cdot10^7\ \dfrac{m}{s}.$$

(b) Let's use the relationship between distance, speed and time for both galaxies:

$$\Delta t=\dfrac{r}{v}=\dfrac{r}{rH}=\dfrac{1}{H},$$ $$\Delta t=\dfrac{1}{1.58\cdot10^{-18}\ s^{-1}}=6.33\cdot10^{17}\ s,$$ $$\Delta t=6.33\cdot10^{17}\ s\cdot\dfrac{1\ h}{3600\ s}\cdot\dfrac{1\ day}{24\ h}\cdot\dfrac{1\ yr}{365\ days}=20\cdot10^9\ yr.$$