Determining the Distance between Lightning and an Observer

We can actually measure the distance between the location of lightning and an observer according to his observation. Let's illustrate that by treating the question in a book by Paul Hewitt.

What is the approximate distance of a thunderstorm when you note a 3-s delay between the flash of lightning and the sound of thunder? 

To make things easier, let's assume that the speed of light is 300,000,000 m/s and that the speed of sound is about 340 m/s (speed of sound in air at 20 deg Celsius). Indeed, light travels faster than sound. Consequently, when light and sound came from the same source, light arrives to a certain destination first.

Variables and Relationships
Let

$latex \displaystyle t_{L}$ be the time for light to travel the distance between the origin of lighning and you.

and

$latex \displaystyle t_{S}$ be the time for sound to travel the same distance.

From the question, 3-s delay between the flash of lightning and the sound of thunder, we can translate this into a mathematical equation. 

$latex \displaystyle t_{S}=t_{L}+ 3 \textup{ s} \longrightarrow (\textup{Eq.} 1) $ 

Let's call this as Eq. 1.

Both travel at constant speeds. So, the relationship among the speed, distance travelled and time can be expressed mathematically as

$latex v_{S}=\frac{d}{t_{S}} \longrightarrow (\textup{Eq.} 2) $

$latex v_{L}=\frac{d}{t_{L}} \longrightarrow (\textup{Eq.} 3) $

Solution

We substitute the expressions for the times described by Eq. 2 and Eq. 3 into Eq. 1 and we'll have.

$latex \frac{d}{v_{S}} = \frac{d}{v_{L}} + 3 \textup{ s} $

Now, we have one equation with only one unknown, that is, d. 

$latex \frac{d}{v_{S}} - \frac{d}{v_{L}} = 3 \textup{ s} $

$latex d \Bigg (\frac{1}{v_{S}} - \frac{1}{v_{L}} \Bigg) = 3 \textup{ s} $

$latex d = \frac {3 \textup{ s}}{\cfrac{1}{v_{S}} - \cfrac{1}{v_{L}} }$
$latex d = \frac {3 \textup{ s}}{\cfrac{1}{340 \textup{ m/s}} - \cfrac{1}{300,000,000 \textup{ m/s}} }$
$latex d = 1,020 \textup{ m } $
And that's the distance we're looking for. Well, this is just an estimate. Here we've rounded off the value of speed of light. In reality it's not really exactly 300,000,000 m/s. Also, the sound may not really be 340 m/s but may depend on the temperature of the air in the surrounding. But anyway, the important thing here is the process in solving the question and not the values. :-)