Great Ideas of Mathematics
Activity 2.2.5
Sept 2011
Galileo’s Odd Number Pattern
In studying the motion of falling bodies, Galileo did experiments with round objects rolling down
ramps. He discovered a curious pattern that reappeared in all his experiments. If you make
measurements at equal time intervals (and starting when the object just begins to roll), the distances
traveled increase in the pattern of odd numbers. For example, say you mark the position of a rolling
ball every 2 seconds (this can be done with a timed camera). If it rolls 4 feet in the first two seconds,
then it will roll 4 × 3 = 12 feet in the next two seconds, 4×5 = 20 feet in the next two seconds, 4×7 =
28 feet in the next two seconds, and so on. If this pattern is accepted as a given, for any length time
period, it is possible to work out the exact motion of rolling (and by analogy, falling) bodies. This
work sheet will lead you through the development of this idea.
1.
Practice with the odd number pattern. Suppose a ball rolls down a ramp, and travels 9 inches in
the first second. How far will it roll in the second second?
The third?
I __________
I ____________ Using your results, complete table below.
Time period
Distance traveled
during that period
1st sec
2nd sec
3rd sec
4th sec
5th sec
6th sec
9 inch
Now suppose a ball rolls down a different ramp, and travels 1.5 feet in the first 3 seconds. How
far will it roll in the next 3 seconds? ________ The next? ________
complete the table below.
Time period
Distance traveled
during that period
1st 3 sec
I Using your results,
2nd 3 sec 3rd 3 sec 4th 3 sec 5th 3 sec
9 inch
2. In the first problem above, your results should show that during seconds 1 and 2, the ball traveled
a total of 36 inches. Applying the odd number pattern to that result, how far should the ball have
rolled during the next two seconds? That is, during seconds 3 and 4? _______ Does this answer
I
agree with what is already shown in your table for seconds 3 and 4? ______ Repeat the same
steps to for the third 2-second interval. Do your results agree with your original entries in the
table? _______________________
I
Cumulative Distance Traveled. Say a rolling ball travels 5 feet in the first second. What total
distance will it travel in the first 6 seconds? One way to answer this question is to create a table
showing how far the ball rolled in each second, and then add up those results. But we can skip the
table and use our pattern directly. When we total the results for each second we will be adding up
5×1 + 5×3 + 5×5 + 5×7 + 5×9 + 5×11. This can be simplified by factoring out the common 5 from
each term, producing 5(1+3+5+7+9+11). In words, the distance traveled in 6 seconds will be 5 times
the total of the first 6 odd numbers. In a similar way, if a ball travels 2.7 feet in the first 3 minutes,
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Great Ideas of Mathematics
Activity 2.2.5
Sept 2011
then in the first 8 three minute intervals it will travel 2.7 times the sum of the first 8 odd numbers.
3. Say a ball travels 11 inches in the first second. How far will it travel in the first 7 seconds?
What if it travels 5.2 feet in the first 4 seconds. How far will it travel in the first 40 seconds?
(Remember to use 4 second time periods).
4. Adding odd numbers. To utilize the preceding results effectively, it helps to notice a pattern
connected with adding odd numbers. Looking systematically at the sum of the first 2 odds, then
the first 3 odds, then the first 4 odds, and so on, you should see a pattern. Discuss it with your
group members and when you all agree on a pattern record it below.
5. Use your rule to predict the sum of the first 12 odd numbers. Then actually add up the first 12
odd numbers to check if your rule gave the correct answer.
The remaining questions all concern falling objects. It is known from many experiments that when
you drop an object from a height, it will fall 16 feet in the first second. So from now on, assume that
any falling body travels 16 feet in the first second. Also assume that Galileo’s odd number pattern
holds.
6. A sky diver jumps from a plane high in the sky. How far will the sky diver fall in the first 10
seconds?
7. How long does it take the sky diver to fall a total of 500 feet? (You can work this out by trial and
error to the nearest second.
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Great Ideas of Mathematics
Activity 2.2.5
Sept 2011
8. For better accuracy, we can use shorter time intervals than 1 second. Given that a falling body
travels 16 feet in one second, how far must it fall in the first half second? The second half
second? (Hint: if you are unsure how to solve this problem ….. GUESS! That is, guess an
answer, and then, using the odd number pattern for half second intervals, check if your guess
gives the right total distance for the first second of falling.)
9. How far does a falling body travel in the first quarter second after it is dropped?
10. Using similar methods, show that a falling body travels a distance of .16 feet in its first tenth of a
second.
Now we know: a falling body travels .16 feet in the first tenth of a second. We can use this to find
the total distance traveled in amounts of time expressed to the nearest tenth. For example, say you
want to know how far the sky driver falls in the first 3.8 seconds. First we note that 3.8 seconds is the
same as 38 tenths of seconds. (Why?) Next, applying what we saw before, the total distance traveled
in 38 equal time periods will be the distance traveled in the first time period times the sum of the first
38 odd numbers. That is,
Distance traveled = .16×(sum of first 38 odd numbers) = .16×38×38.
This same logic can be used with any other one-decimal-place number in stead of 3.8.
11. How far would a sky diver fall in the first 12.5 seconds?
In 15.8 seconds?
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Great Ideas of Mathematics
Activity 2.2.5
Sept 2011
12. How long would it take the sky diver to fall 500 feet? Give your answer to the nearest .1 seconds.
(You might have to use trial and error again).
Commentary: Just using the patterns of odd numbers and their sums, you were able to work out procedures
for finding the distance a body falls from rest in any whole number of seconds. If you review your results, you
will see that they can be summarized in the following rule:
A body falling from rest for n seconds will travel 16×n×n feet, where n is any whole number.
Then you refined your results to deal with tenths of seconds. Again, if you review your results, you will see
that they can be summarized by the following nearly identical rule.
A body falling from rest for n tenths of seconds will travel 0.16×n×n feet, where n is any whole number.
Similar methods can be applied to deal with hundredths of seconds, or thousandths, or any other subdivision.
When all of these rules are examined together, we find that there is one general rule that applies in all cases:
A body falling from rest for x seconds will travel 16x2 feet, where x is any decimal number.
So, from Gallileo’s modest sounding observation distances traveled in equal time periods increase in the
same proportions as successive odd numbers, we are able to derive a precise formula showing how
distance and travel time are related for a falling body. This is a dramatic example of collecting numerical
data in nature, finding number patterns, and then using our ability to analyze number patterns to achieve
new insights about nature. In the simple examples you have completed, you are retracing the initial steps
that eventually led to calculus. There, essentially the same kind of reasoning, developed to a much more
sophisticated level, lead to unbelievably accurate predictions about all sorts of physical structures and
systems. And the ability to make these predictions is critical to all of our science and technology and
engineering, from computers to space exploration, from I-beams to iphones.
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