Thunder and Lightning

October 26, 2009

Sunday night, as we sat on the patio by the grill, we thought about thunder and lightning. The sky was overcast, and in the distance storms were brewing. Lightning reflected off the clouds above us, followed a short time later by thunder. We started to think about the relationship between the light and sound from the storm. We knew that it would be easy to come up with a quick relationship between the time between the sound of thunder and the flash of lightning.

The light and sound travel the same distance, d. The time the light requires to go from the storm to us is shorter than the time for the sound. The time for the light is t_l = d/c, where c is the speed of light (300,000,000 m/s). The time for sound is t_s = d/v_s, where v_s is the speed of sound (300 m/s). We are interested in the number of seconds between the sound and the light, so let’s compute the time difference using these formula:

t_s – t_l = d/v_s – d/c = d(1/v_s – 1/c)

We can rewrite the quantity (1/v_s – 1/c) = (1/300 – 1/300,000,000) = 1/300 (1-0.000001), which is (to a very good approximation) 1/300.

We arrive at our final relationship:

t_s – t_1 = d/300

where time is in seconds and distance is in meters. So, if we are sitting on our patio and we see the flash of lightning, and then 6 seconds later we hear the thunder, the storm is (6s * 300 m/s) = 1,800 m, or about 1.1 miles.

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