document.write( "Question 844782: A ball is tossed in the air from an initial height of 6 feet, it reaches its maximum height of 10 feet after 3 seconds then comes back down.then it bounces back up to a second maximum height of 1 foot in 1 second
\n" ); document.write( "a) Assume that the motion is parabolic in each case and determine a piecewise function h(t). express all non- exact answers rounded correctly to three decimal places.
\n" ); document.write( "b) now assume that the whole time it is doing this, it is moving in the horizontal(call it the x direction) at a constant speed of 1.5 feet per second. Determine the particles path through space h(x(t)) as a piecewise function as it goes through the initial toss and two bounces leave all non-exact answers rounded to three decimal places.
\n" ); document.write( "guys I need help in home work :) i would appreciate the help \n" ); document.write( "

Algebra.Com's Answer #508947 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
This problem reads as a math problem, so I will forget that I know physics.
\n" ); document.write( "a) Assuming that the motion is parabolic, we expect the pieces of the graph of $\"h%28t%29\"$ as a function of $\"t\"$ to look like this:
\n" ); document.write( "

The first piece has axis of symmetry $\"t=3\"$ seconds,
\n" ); document.write( "apex/vertex at $\"h=10\"$ feet,
\n" ); document.write( "and $\"h%280%29=6\"$ feet.
\n" ); document.write( "From $\"f%28t%29=t%5E2\"$ , a flip, a vertical stretch, and a translation bring us to
\n" ); document.write( " $\"h%28t%29=10%2BK%28t-3%29%5E2\"$ as a candidate function.
\n" ); document.write( "That function graphs as a parabola with a vertex at $\"t=3\"$ seconds with $\"h=10\"$ feet.
\n" ); document.write( "We know that we need $\"K%3C0\"$ for the vertex to be a maximum, and that we need $\"h%280%29=6\"$ feet.
\n" ); document.write( " $\"h%280%29=10%2BK%280-3%29%5E2=6\"$
\n" ); document.write( " $\"10%2BK%2A3%5E2=6\"$
\n" ); document.write( " $\"10%2B9K=6\"$
\n" ); document.write( " $\"9K=6-10\"$
\n" ); document.write( " $\"9K=-4\"$
\n" ); document.write( " $\"K=-4%2F9\"$
\n" ); document.write( "So the first piece of the function, from $\"t=0\"$ to the first bounce point, where $\"h%28t%29=0\"$ is
\n" ); document.write( " $\"h%28t%29=10-%284%2F9%29%28t-3%29%5E2\"$
\n" ); document.write( " $\"h%28t%29=0\"$ --> $\"10-%284%2F9%29%28t-3%29%5E2=0\"$ --> $\"10=%284%2F9%29%28t-3%29%5E2\"$ --> $\"10%2A%289%2F4%29=%28t-3%29%5E2\"$ --> $\"t-3=sqrt%2810%2A%289%2F4%29%29\"$ --> $\"t-3=%283%2F2%29%2Asqrt%2810%29\"$ --> $\"t=3%2B%283%2F2%29%2Asqrt%2810%29\"$ seconds = approx. $\"7.743\"$ seconds .
\n" ); document.write( "So the first piece is
\n" ); document.write( " $\"highlight%28h%28t%29=10-%284%2F9%29%28t-3%29%5E2%29\"$ } for $\"highlight%280%3C=t%3C7.743%29\"$ seconds .
\n" ); document.write( "
\n" ); document.write( "The second bounce takes the ball up to $\"h=1\"$ foot 1 second later,at
\n" ); document.write( " $\"t=4%2B%283%2F2%29%2Asqrt%2810%29=about8.743\"$ seconds, and parabola symmetry requires that the ball get back to $\"h=0\"$ in another second at $\"t=5%2B%283%2F2%29%2Asqrt%2810%29\"$ seconds.
\n" ); document.write( "The function $\"y=1-t%5E2\"$ goes from $\"y%28-1%29=0\"$ to a maximum of $\"y%280%29=1\"$ in 1 second.
\n" ); document.write( "We need the same shape, but shifted right so the vertex is at $\"t=4%2B%283%2F2%29%2Asqrt%2810%29=8.743\"$ instead of $\"t=0\"$ .
\n" ); document.write( "So $\"highlight%28h%28t%29=1-%28t-8.743%29%5E2%29\"$ for $\"highlight%287.743%3C=t%3C=9.743%29\"$ would cover the second piece of the piecewise function.
\n" ); document.write( "
\n" ); document.write( "b) $\"x%28t%29=1.5t\"$ <--> $\"x%28t%29=%283%2F2%29%2At\"$ <--> $\"t=%282%2F3%29x\"$
\n" ); document.write( "We can express the trajectory function $\"h%28x%29\"$ by substituting $\"t=%282%2F3%29x\"$ into $\"h%28t%29\"$ .
\n" ); document.write( "The $\"t%3C=0%3C7.743\"$ part of the piecewiese function corresponds to $\"0%3C=x%3C7.743%2A1.5\"$ <--> $\"highlight%280%3C=x%3C11.615%29\"$
\n" ); document.write( "

\n" ); document.write( "and we get $\"h=highlight%2810-%284%2F81%29%282x-9%29%5E2%29\"$ .
\n" ); document.write( "For the second piece, $\"t%3C=0%3C7.743\"$ corresponds to
\n" ); document.write( " $\"11.615%3C=x%3C=9.743%2A1.5\"$ <--> $\"11.615%3C=x%3C=14.615\"$
\n" ); document.write( "and we get
\n" ); document.write( "

\n" ); document.write( "However, using the more accurate $\"4%2B%283%2F2%29%2Asqrt%2810%29=8.743416\"$ value, we get
\n" ); document.write( " $\"3%2A8.743416=26.230\"$ when rounded to 3 decimal places.
\n" ); document.write( "So $\"h=highlight%281-%282x-26.230%29%5E2%2F9%29\"$ for $\"highlight%2811.615%3C=x%3C=14.615%29\"$ .
\n" ); document.write( "
\n" ); document.write( "IN REVENGE
\n" ); document.write( "for making me calculate a bunch of ugly numbers,
\n" ); document.write( "I want to point out that the problem is really out of this world.
\n" ); document.write( "The acceleration of the ball after it was tossed in the air was
\n" ); document.write( " $\"2%2A%284%2F9%29=8%2F9\"$ $\"feet%2Fsecond%5E2\"$ , but on my planet the acceleration of gravity is $\"32\"$ $\"feet%2Fsecond%5E2\"$ .
\n" ); document.write( "On earth's moon, it would be $\"5.3\"$ $\"feet%2Fsecond%5E2\"$ .
\n" ); document.write( "That ball is being tossed on a quite small satellite or asteroid, with little gravity.
\n" ); document.write( "after it bounces, the acceleration suddenly increases to
\n" ); document.write( " $\"2\"$ $\"feet%2Fsecond%5E2\"$ .
\n" ); document.write( "That is even stranger, because gravity should remain constant on any planet, satellite, or asteroid.
\n" ); document.write( "It would have been more consistent if the bounce had made the ball reach 1 foot is 1.5 seconds. Then the acceleration of gravity before and after the bounce would have been the same. \n" ); document.write( "

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