Question

How many light years away is Mars from Earth?

Answer 1

Hint: We need to understand the unit of measurement of length that is given in the problem. We need to find the relation between the light years and the common units of measurement such as the meters and the kilometers to solve this problem.

Complete Solution:
We know that the earth and the mars are the planets in the solar system. We usually use the kilometers as the unit for measuring the distances between the sun to the planets, the satellites and between the planets as the distance is not large enough to be expressed in large quantities as light years.

Now, let us understand what a light year exactly is. The light year is the distance travelled by light through vacuum in an earth year time. We know that an earth year is 365 earth days. We can find the total time in seconds that constitute a year as –
$$\begin{array}{rl}& 1\text{ year = 365days}\times \text{24hrs}\times 60min\times 60s\\ & \therefore 1yr=3.1536\times {10}^{7}s\end{array}$$

Now, we can find the distance travelled by the light at the speed ‘c’ in this time which will give us the value of 1 light year as –
$$\begin{array}{rl}& c=3\times {10}^{8}m{s}^{-1}\\ & \Rightarrow \text{1light year = Speed }\times \text{ time}\\ & \Rightarrow 1ly=3\times {10}^{8}m{s}^{-1}\times 3.1536\times {10}^{7}s\\ & \therefore 1ly=9.4608\times {10}^{15}m\end{array}$$

So, we get the value of 1 light year to be in the fifteenth power of ten which is very large. We can use this relation to find the distance between the earth and mars in light years.

The distance from mars to earth is 113.64 million km. this can be converted to light years as –
$$\begin{array}{rl}& \text{Distance from earth to mars = 1}\text{.1364}\times \text{1}{\text{0}}^{11}\text{m}\\ & \Rightarrow \text{In light years, Distance = }{\displaystyle \frac{\text{1}\text{.1364}\times \text{1}{\text{0}}^{11}\text{m}}{9.4608\times {10}^{15}m}}\\ & \therefore \text{Distance between earth and mars = 1}\text{.2012}\times \text{1}{\text{0}}^{-5}ly\end{array}$$

This is the required solution.

Note:
We can see that the light year is an enormous unit to measure the distance between planets as the value turns out to be so small. We prefer the usual units of measurement like the kilometers to measure the inter-solar distances which is more meaningful.