Today is 19 January 2011 -- How long does sound take to travel?

Today is 19 January 2011 -- How long does sound take to travel?

The volcano Krakatoa erupted 27 August 1883 with a sound so loud it was heard 3,000 miles away, on Rodriguez Island. You know the speed of sound (It was given in one of the video presentations as 1100 ft/sec.) Although the speed of sound changes based on air density and temperature, we can assume the speed to be the same as under normal atmospheric conditions. How do you solve this?

The equation is still S = r x t, where S = distance, r = speed or velocity, and t = time. Rearrange the equation to solve for time: S/r = t. Then substitute the known information: 3,000 mi/1100 ft per sec = t. You need to convert miles to feet or feet to miles to have the same units. Since students were taught there are 5,280 feet per mile, we can write:

[(3,000 miles)(5280 feet/mile)] / (1100 feet/sec) = time.

In class I taught the students to use scientific notation to solve the problem because it is easier. I also reminded them to cancel units where possible and to remember that the denominator "goes into the numerator," not the other way around. Here are the steps:

[(3,000 miles)(5280 feet/mile)] / (1100 feet/sec) = time.

(3.0 x 10³ mile)(5.28 x 10³ feet /mile) 15.84 x 10⁶

-------------------------------------------- = --------------

(1.1 x 10³ feet/sec) 1.1 x 10³/sec

Divide 15.84 x 10⁶ (numerator) by 1.1 x 10³ (denominator). Work first with the real numbers:

_____

1.1 ) 15.84

Move the decimal point of the divisor (1.1) over to the right one place until it is a whole number. What you do “outside,” you have to do “inside” too:

_____

11) 158.4

Now, ask yourself, “Does 11 go into 1?” No. So then ask, “Does 11 go into 15?” Yes, once. So write a “1” on the line over the “5,” multiply 1 x 11, write the answer below the 15, then subtract from 15:

__1___

11) 158.4

11

4

Then, bring down the “8” and ask how many times 11 goes into 48. It goes 4 times, so you then multiply 11 x 4 on the next line and write “44.” Subtract 44 from 48 to get “4.”

__14___

11) 158.4

11

48

44

4

Bring down the last digit, “4” and divide 11 into another 44:

__14.4___

11) 158.4

11

48

44

44

44

You have no remainder so you are finished. Remember to place a decimal point above the one inside the box.

Next, handle the exponential term. When you multiply exponentials, you add them; when you divide, you subtract:

10⁶ / 10³ means 10^{(6 – 3)} = 10³

So your answer is 14.4 x 10³ seconds. How did the unit “seconds” get into the numerator? It is because it was the denominator of a denominator. When you divide, you invert the denominator and multiply, which puts the seconds into the numerator.

Finally, how many seconds are there in one hour? There are 3600 or in scientific format, 3.6 x 10³ sec/hr. As we did before, divide 3.6 into 14.4. What is your answer? I got “4 hours.” Remember, the exponential terms will cancel out as will the unit “seconds.”

The people on Rodriguez Island reported hearing sounds like the “roar of canons” on the night of 26 August 1883. It took 4 hours for the sound of Krakatoa erupting to reach them. By the way, how can it be that the eruption happened on 27 August and they heard it the night of the 26^th?