document.write( "Question 885774: A stream of water in steady flow from a kitchen faucet. At the faucet the diameter of stream is 0.96 cm. The stream fills a 125 cm^3 in 16.3 seconds. Find the diameter of the stream 13 cm below the opening of the faucet. \n" ); document.write( "

Algebra.Com's Answer #535424 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
Disclaimer/apology:
\n" ); document.write( "I apologize to you in advance for possible mistakes. I try to keep all my units straight, and calculate correctly, but I am prone to punching in the wrong numbers in the computer and calculator/
\n" ); document.write( "I also apologize to professor Walter Lewin.
\n" ); document.write( "My grasp of physics was never strong enough for my taste, even after paying attention at some (though not all) of his lectures.
\n" ); document.write( "If you do not like my answer ask again.
\n" ); document.write( "If this website will not produce a suitable answer, maybe the forum at artofproblemsolving.com will (they have Physics sections).\r
\n" ); document.write( "\n" ); document.write( "The flow rate in $\"cm%5E3%2Fsecond\"$ is
\n" ); document.write( " $\"125%2F16.3\"$ .
\n" ); document.write( "Assuming a perfectly circular cross section, with $\"diameter=0.96cm\"$ --> $\"radius=0.48cm\"$ ,
\n" ); document.write( "the cross section of the stream at the faucet (in $\"cm%5E2\"$ ) is
\n" ); document.write( " $\"pi%2A0.48%5E2\"$
\n" ); document.write( "The linear velocity of the water at the faucet (in cm/second) is
\n" ); document.write( " $\"%28%28125%2F16.3%29%29%2F%28pi%2A0.48%5E2%29=about+10.6\"$ .
\n" ); document.write( "Under no other force but gravity, that linear velocity would increase with time.
\n" ); document.write( "Let $\"t\"$ = time in seconds from the moment the water leaves the faucet.
\n" ); document.write( " $\"v%28t%29\"$ = linear velocity of the water (in cm/second) at $\"t\"$ seconds.
\n" ); document.write( " $\"d%28t%29\"$ = distance between the water and the faucet (in cm) at $\"t\"$ seconds.
\n" ); document.write( "Let's take the acceleration of gravity as $\"g=9.8\"$ $\"m%2Fsecond%5E2=980\"$ $\"cm%2Fsecond%5E2\"$ .
\n" ); document.write( "Then, $\"v%28t%29=10.6%2B980t\"$ and $\"d%28t%29=10.6t%2B980t%5E2%2F2=10.6t%2B490t%5E2\"$
\n" ); document.write( "When $\"d%28t%29=13\"$ $\"10.6t%2B490t%5E2=13\"$ <---> $\"10.6t%2B490t%5E2-13=0\"$
\n" ); document.write( "That quadratic equation would have two solutions, but we are only interested in the positive one.
\n" ); document.write( "So we calculate $\"t=%28-10.6+%2B-+sqrt%2810.6%5E2-4%2A490%2A%28-13%29%29%29%2F%282%2A490%29\"$ = approximately $\"149.4%2F980=about0.152\"$ seconds.
\n" ); document.write( "Then, at that time, water has traveled 13 cm from the faucet and it is moving at a velocity (in cm/s) of
\n" ); document.write( " $\"v%280.1542%29=10.6%2B980t=10.6%2B149.4=160\"$
\n" ); document.write( "The flow rate $\"cm%5E3%2Fsecond\"$ at that point (13 cm from the faucet) is
\n" ); document.write( "the same as before, and is equal to
\n" ); document.write( "the the linear velocity times the cross section of the stream at that point.
\n" ); document.write( "From the flow rate, we could calculate the cross section radius and diameter.
\n" ); document.write( "However, it is easier to calculate the diameter knowing that the linear velocity is inversely proportional to the square of the diameter.
\n" ); document.write( "The linear velocity increased by a factor of $\"160%2F10.6=about15.09\"$ ,
\n" ); document.write( "so the diameter must have decreased by a factor of $\"sqrt%2810.09%29=about3.9\"$ .
\n" ); document.write( "So the diameter would be $\"0.96cm%2F3.9=0.25cm\"$ .
\n" ); document.write( "The calculation may not quite agree with reality because other physical influences and phenomena may be at work. \n" ); document.write( "

\n" );