document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=565316>Question 565316</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Reading \"50 Mathematical Ideas you really need to know\".  It gives Galileo's distance formula - y = 16 times x squared.  It says the factor 16 is because feet and seconds are the chosen measurement units.  Don't understand why 16?  This is the introduction to Calculus and I haven't got past this number 16 yet.\r<BR>\n" );
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document.write( "Thank you, Chris. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #365768 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=365768\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the formula is 16 * x^2 because the measurement was done in feet and seconds.<BR>\n" );
document.write( "if the measurement was done in some other measure, like inches in half a second intervals, the factors would have been different.<BR>\n" );
document.write( "the more generalized formula would be:<BR>\n" );
document.write( "d = k * x^2 where:<BR>\n" );
document.write( "k is the distance the object falls in 1 unit of time.<BR>\n" );
document.write( "x is the unit of time.<BR>\n" );
document.write( "for example:<BR>\n" );
document.write( "assume the distance was in inches and the time in seconds.<BR>\n" );
document.write( "the formula would then have become:<BR>\n" );
document.write( "d = (16*12 = 192) * x^2 where:<BR>\n" );
document.write( "192 is the distance in inches and x is the number of seconds.<BR>\n" );
document.write( "assume the distance was in feet and the time interval was half a second.<BR>\n" );
document.write( "the formula would then have become:<BR>\n" );
document.write( "d = 4 * x^2 where:<BR>\n" );
document.write( "4 is the number of feet the object fell in half a second and x is the number of half seconds.<BR>\n" );
document.write( "the formula works the same either way.<BR>\n" );
document.write( "the following chart should demonstrate that.<BR>\n" );
document.write( "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.<BR>\n" );
document.write( "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.<BR>\n" );
document.write( "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.<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "seconds     half seconds        a              b             c\r\n" );
document.write( "1               2               16             192           16\r\n" );
document.write( "2               4               64             768           64\r\n" );
document.write( "3               6               144            1728          144\r\n" );
document.write( "15              30              3600           43200         3600\r\n" );
document.write( "</PRE><BR>\n" );
document.write( "note that the results are the same regardless of the units used as long as you did the conversions correctly.<BR>\n" );
document.write( "3600 feet * 12 inches per foot equals 43200 inches.<BR>\n" );
document.write( "16 * 15^2 = 3600<BR>\n" );
document.write( "4 * 30^2 = 3600<BR>\n" );
document.write( "the formula is valid regardless of the units used.<BR>\n" );
document.write( "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.<BR>\n" );
document.write( "from what i read, he may have actually used meters.<BR>\n" );
document.write( "the formula would have been d = 4.9 * x^2 where 4.9 is the number of meters in 1 second.<BR>\n" );
document.write( "4.9 meters is roughly equivalent to 16 feet which is why you see 16 feet.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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