Question

If Hubble’s law can be extrapolated to very large distances, at what distance would the apparent recessional speed become equal to the speed of light ?

Short answer

The distance at which the apparent recessional speed become equal to the speed of light is $1.4\times {10}^{10}\text{\hspace{0.17em}ly}$.

Step-by-step solution

Explain Hubble’s law and determine the formula

Hubble’s law connects the recession speed $v$of a galaxy and its distance $r$from us, are directly proportional.

$v=Hr$ …… (1)

Here,$H$ is the Hubble constant.

Find the distance

Consider the Hubble’s law from the Equation (1) and the speed of the light $c$as follows:

$\begin{array}{c}v=Hr\\ =c\end{array}$ ...... (2)

Rewrite the equation (2) as follows and substitute the constant values of and .

$\begin{array}{l}r=\frac{c}{H}\\ r=\frac{3.0\times {10}^8\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.}{0.0218\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\text{.ly}}\\ r=1.376\times {10}^{10}\text{\hspace{0.17em}ly}\\ r\approx \text{1.4}\times {\text{10}}^{10}\text{\hspace{0.17em}ly}\end{array}$

Therefore,thedistance, at which the apparent recessional speed become equal to the speed of light is.$1.4\times {10}^{10}\text{\hspace{0.17em}ly}$