the formula is 16 * x^2 because the measurement was done in feet and seconds.
if the measurement was done in some other measure, like inches in half a second intervals, the factors would have been different.
the more generalized formula would be:
d = k * x^2 where:
k is the distance the object falls in 1 unit of time.
x is the unit of time.
for example:
assume the distance was in inches and the time in seconds.
the formula would then have become:
d = (16*12 = 192) * x^2 where:
192 is the distance in inches and x is the number of seconds.
assume the distance was in feet and the time interval was half a second.
the formula would then have become:
d = 4 * x^2 where:
4 is the number of feet the object fell in half a second and x is the number of half seconds.
the formula works the same either way.
the following chart should demonstrate that.
a is from the equation d = 16 * x^2 where 16 is the the number of feet in 1 second and x is the number of seconds.
b is from the equation d = 192 * x^2 where 192 is the number of inches in 1 second and x is the number of seconds.
c is from the equation d = 4 * x^2 where 4 is the number of feet in half a second and x is the number of half seconds.
seconds half seconds a b c
1 2 16 192 16
2 4 64 768 64
3 6 144 1728 144
15 30 3600 43200 3600
note that the results are the same regardless of the units used as long as you did the conversions correctly.
3600 feet * 12 inches per foot equals 43200 inches.
16 * 15^2 = 3600
4 * 30^2 = 3600
the formula is valid regardless of the units used.
the reason why the formula you are looking at is d = 16 * x^2 where 16 is the number of feet it dropped in 1 second and x is the number of seconds is because those are the units of measure that were used in galileo's observation.
from what i read, he may have actually used meters.
the formula would have been d = 4.9 * x^2 where 4.9 is the number of meters in 1 second.
4.9 meters is roughly equivalent to 16 feet which is why you see 16 feet.