Question

The distance of a galaxy from the Earth is of the order of ${{10}^{25}}m$. Calculate the order of magnitude of the time taken by light to reach us from the galaxy.

Answer 1

Distance is defined as the product of speed and time. Speed of light is of the order of ${{10}^{8}}m{{s}^{-1}}$. Time taken by light can easily be calculated from the distance traveled by light from the galaxy to the Earth and the speed of light in air.
Formula used:
$t=\dfrac{d}{s}$

Complete step-by-step solution:
We know that distance traveled by an object is equal to the product of the speed of the object and the time taken by the object to cover this distance. Mathematically, distance traveled is given by
$d=st$
where
$d$ is the distance traveled by an object
$s$ is the speed of the object
$t$ is the time taken by the object to cover the distance $d$
Let this be equation 1.
From equation 1, it is clear that time taken by an object to cover a given distance is equal to the ratio of distance covered by the object to the speed of the object. Therefore, equation 1 can be rewritten as
$t=\dfrac{d}{s}$
Let this be equation 2.
Coming to our question, we are provided that distance traveled by light from a galaxy to the Earth is of the order of ${{10}^{25}}m$. We are required to determine the order of time taken by light to cover this distance.
We know that the speed of light in air is given by
${{s}_{light}}=3\times {{10}^{8}}m{{s}^{-1}}$
where
${{s}_{light}}$ is the speed of light in air
Let this be equation 3.
If distance travelled by light from the given galaxy to the Earth and the time taken by light to cover this distance are denoted as ${{d}_{light}}$ and ${{t}_{light}}$, respectively, then, using equation 2, ${{t}_{light}}$ is given by
${{t}_{light}}=\dfrac{{{d}_{light}}}{{{s}_{light}}}=\dfrac{{{10}^{25}}}{3\times {{10}^{8}}m{{s}^{-1}}}=0.33\times {{10}^{17}}=3.3\times {{10}^{16}}s$
Let this be equation 4.
Clearly, from equation 4, time taken by light to travel from the given galaxy to the Earth is of the order of ${{10}^{16}}s$.

Note: Students need to understand equation 4 clearly. Here, we got the answer as $0.33\times {{10}^{17}}s$ at first. While taking the order of magnitude of time taken, it should be kept in mind that the answer is in the form
$x.yz\times {{10}^{n}}$
where
it is necessary that $x > 0$
Therefore, $0.33\times {{10}^{17}}s$ is changed to $3.3\times {{10}^{16}}s$, to arrive at the correct order of magnitude of time taken by light to travel to the Earth.