document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=118531>Question 118531</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A flare is fired as a stress signal and its height ,h in metres above the ground, t seconds after firing, is provided in the table below. Algebraically determine the quadratic function that defines the height of the flare above the ground t seconds after firing, and use it to determine the flare's height at 2.4 seconds.\r<BR>\n" );
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document.write( "t---0 1   2   3   4  <BR>\n" );
document.write( "h---2 97 182 257 322<BR>\n" );
document.write( "I tried y=-2ak^2<BR>\n" );
document.write( "97=-2a(1)^2<BR>\n" );
document.write( "97/-2=-2a/-2<BR>\n" );
document.write( "a=-48.5  and was completly lost :( Please Help Me :) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #86643 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=86643\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The general formula for the distance traveled by a moving object that has an initial velocity, initial distance (height in this case), and being acted upon by a constant force is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=%281%2F2%29at%5E2%2Bv%5B0%5Dt%2Bh%5B0%5D ALIGN=MIDDLE  ALT=\"h=%281%2F2%29at%5E2%2Bv%5B0%5Dt%2Bh%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the acceleration due to the constant force,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5B0%5D ALIGN=MIDDLE  ALT=\"v%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the initial velocity, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5B0%5D ALIGN=MIDDLE  ALT=\"h%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the initial distance (height).\r<BR>\n" );
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document.write( "Since we can presume that the flare doesn't have any continuous propulsion, i.e. the initial velocity was provided by a controlled explosion in the chamber of the gun that fired it, we can assume that the only force acting upon the flare is gravity.  The acceleration due to the earth's gravity is approximately <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g=9.8%2A%28m%2Fs%5E2%29 ALIGN=MIDDLE  ALT=\"g=9.8%2A%28m%2Fs%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We also know that the initial height (height at time 0) is 2 meters because that is given in the tabular data.\r<BR>\n" );
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document.write( "Now the problem is to determine the initial velocity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5B0%5D ALIGN=MIDDLE  ALT=\"v%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the tabular data.  First a bit of discussion about the velocity.  Remember, velocity is a vector quantity, i.e. it has both a magnitude and a direction.  We could assume that the flare was fired straight up, in which case we need not be concerned with the magnitude of the velocity as the flare exits the muzzle of the flare gun as compared to the vertical component of that velocity because they would be the same.  However, it turns out that it doesn't matter what angle to the vertical that the gun is pointed when it is fired because we can calculate the vertical component of the velocity directly.\r<BR>\n" );
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document.write( "If the flare gun had been fired in a zero-gravity environment, the vertical component (whatever that means in a zero-gravity environment) of velocity of the flare would never change from the initial vertical component, c.f. Newton's Second Law of Motion.  In our case, the actual height achieved is smaller by the distance given by the gravitational acceleration term of the distance formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29gt%5E2 ALIGN=MIDDLE  ALT=\"%281%2F2%29gt%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Note that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g ALIGN=MIDDLE  ALT=\"g\"   DISABLED_style= \"max-width:90%; height:auto;\" > in this situation is a negative quantity presuming we assign the positive direction to the initial velocity.\r<BR>\n" );
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document.write( "We know from the tabular data that the height of the flare after 4 seconds is 322 meters.  Had we fired the flare in a zero-gravity environment, the height of the flare would have been <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=322+%2B+%281%2F2%29%2A9.8%2A%284%29%5E2 ALIGN=MIDDLE  ALT=\"322+%2B+%281%2F2%29%2A9.8%2A%284%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or approximately 400.4 meters -- meaning that the projectile would have traveled 400.4 minus the initial height of 2 meters or 398.4 meters in that 4 seconds.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=398.4%2F4=99.6 ALIGN=MIDDLE  ALT=\"398.4%2F4=99.6\"   DISABLED_style= \"max-width:90%; height:auto;\" > meters/sec is then the initial velocity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5B0%5D ALIGN=MIDDLE  ALT=\"v%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now we have sufficient information to create the quadratic statement that expresses height as a function of time:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=%281%2F2%29%2A-9.8%2At%5E2%2B99.6t%2B2 ALIGN=MIDDLE  ALT=\"h%28t%29=%281%2F2%29%2A-9.8%2At%5E2%2B99.6t%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In order to determine the height at 2.4 seconds, evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%282.4%29 ALIGN=MIDDLE  ALT=\"h%282.4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I'll leave the arithmetic for you to complete.\r<BR>\n" );
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document.write( "Hope that helps.<BR>\n" );
document.write( "John\r<BR>\n" );
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document.write( "P.S.<BR>\n" );
document.write( "Since the height data provided was expressed to the nearest meter, the appropriate precision for your answer is also to the nearest meter.  For example, if your calculator tells you the answer is something like 212.8160 meters, you should round it to 213 meters.  The lesson is: Never give an answer more precise than the least precise measurement in the given information.\r<BR>\n" );
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document.write( "J. </SPAN>\n" );
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