document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=72917>Question 72917</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose a baseball is shot up from the ground straight up with an initial velocity of 64 feet per second. A function can be created by expressing distance above the ground, s, as a function of time, t. This function is s = -16t2 + v0t + s0 \r<BR>\n" );
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document.write( "•16 represents 1/2g, the gravitational pull due to gravity (measured in feet per second2). \r<BR>\n" );
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document.write( "•v0 is the initial velocity (how hard do you throw the object, measured in feet per second).\r<BR>\n" );
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document.write( "•s0 is the initial distance above ground (in feet). If you are standing on the ground, then s0 = 0.\r<BR>\n" );
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document.write( "a)What is the function that describes this problem?\r<BR>\n" );
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document.write( "Answer: \r<BR>\n" );
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document.write( "b) The ball will be how high above the ground after 1 second?<BR>\n" );
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document.write( "c) How long will it take to hit the ground?<BR>\n" );
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document.write( "d) What is the maximum height of the ball?  What time will the maximum height be attained?\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "Show work in this space.	\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #52300 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=52300\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Suppose a baseball is shot up from the ground straight up with an initial velocity of <BR>\n" );
document.write( "64 feet per second. A function can be created by expressing distance above the ground, s, <BR>\n" );
document.write( "as a function of time, t. This function is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+-16t%5E2+%2B+v%5Bo%5Dt+%2B+s%5Bo%5D ALIGN=MIDDLE  ALT=\"s+=+-16t%5E2+%2B+v%5Bo%5Dt+%2B+s%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "•16 represents <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%282%2Ag%29 ALIGN=MIDDLE  ALT=\"1%2F%282%2Ag%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the gravitational pull due to gravity (measured in feet per second^2). \r<BR>\n" );
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document.write( "•<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5Bo%5D ALIGN=MIDDLE  ALT=\"v%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the initial velocity (how hard do you throw the object, measured in feet per second).\r<BR>\n" );
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document.write( "•<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5Bo%5D ALIGN=MIDDLE  ALT=\"s%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the initial distance above ground (in feet). If you are standing on the ground, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+s%5Bo%5D=+0 ALIGN=MIDDLE  ALT=\"+s%5Bo%5D=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "a)What is the function that describes this problem?<BR>\n" );
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document.write( "Answer: <BR>\n" );
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document.write( "Start with the equation you are given:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+-16t%5E2+%2B+v%5Bo%5Dt+%2B+s%5Bo%5D ALIGN=MIDDLE  ALT=\"s+=+-16t%5E2+%2B+v%5Bo%5Dt+%2B+s%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You are given that the initial velocity is +64 feet per second.  Substitute that value for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5Bo%5D ALIGN=MIDDLE  ALT=\"v%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "and the equation then becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+-16t%5E2+%2B+64t+%2B+s%5Bo%5D ALIGN=MIDDLE  ALT=\"s+=+-16t%5E2+%2B+64t+%2B+s%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "Since the baseball is shot up from ground level, you know that the starting height is<BR>\n" );
document.write( "zero.  Therefore, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5Bo%5D ALIGN=MIDDLE  ALT=\"s%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is zero and it just drops from the equation.  You are left with:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+-16t%5E2+%2B+64t ALIGN=MIDDLE  ALT=\"s+=+-16t%5E2+%2B+64t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and that is the answer to this part of the problem.<BR>\n" );
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document.write( "b) The ball will be how high above the ground after 1 second?<BR>\n" );
document.write( "Answer:	<BR>\n" );
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document.write( "Show work in this space.	<BR>\n" );
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document.write( "In the equation that we got for part a) of this problem, t is in seconds.  So all we<BR>\n" );
document.write( "have to do is put 1 second into the equation in place of t.  If you do that you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+-16%2A%281%29%5E2+%2B+64%2A1 ALIGN=MIDDLE  ALT=\"s+=+-16%2A%281%29%5E2+%2B+64%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This reduces to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+-16+%2B+64+=+48 ALIGN=MIDDLE  ALT=\"s+=+-16+%2B+64+=+48\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So the baseball is 48 feet above the ground at 1 second after it is launched.<BR>\n" );
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document.write( "c) How long will it take to hit the ground?<BR>\n" );
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document.write( "Show work in this space. <BR>\n" );
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document.write( "There are two times that the ball is at a height of zero feet.  The first time is just<BR>\n" );
document.write( "at the instant launch begins and the second time is when it hits the ground after it<BR>\n" );
document.write( "falls to Earth.<BR>\n" );
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document.write( "Return to the equation we developed as the answer to part a) of this problem.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+-16t%5E2+%2B+64t ALIGN=MIDDLE  ALT=\"s+=+-16t%5E2+%2B+64t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Set s equal to zero because we are interested in finding when the baseball is at ground level.<BR>\n" );
document.write( "With s set as being equal to zero the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+-16t%5E2+%2B+64t ALIGN=MIDDLE  ALT=\"0+=+-16t%5E2+%2B+64t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Factor a t out of the right side of the equation to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+t%2A%28-16t+%2B+64%29 ALIGN=MIDDLE  ALT=\"0+=+t%2A%28-16t+%2B+64%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Notice that this equation will be true whenever one of the factors on the right side is zero.<BR>\n" );
document.write( "When one of these factors is zero, the whole right side multiplies out to be zero and<BR>\n" );
document.write( "is then equal to the left side.<BR>\n" );
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document.write( "So this equation is satisfied if either t = 0 or if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-16t+%2B+64%29+=+0 ALIGN=MIDDLE  ALT=\"%28-16t+%2B+64%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Well we knew already that at t = 0 (just at the instant of launch) the ball was at ground<BR>\n" );
document.write( "level.  So we need to solve the equation for the second factor to find out the second time<BR>\n" );
document.write( "that the ball is at ground level. Therefore solve the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-16t+%2B+64%29+=+0 ALIGN=MIDDLE  ALT=\"%28-16t+%2B+64%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Subtract 64 from both sides to eliminate the 64 on the right side. The equation then becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-16t+=+-64 ALIGN=MIDDLE  ALT=\"-16t+=+-64\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "If you then divide both sides by -16 the answer for t becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+4 ALIGN=MIDDLE  ALT=\"t+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So the ball is in the air a total of 4 seconds.<BR>\n" );
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document.write( "d) What is the maximum height of the ball?  What time will the maximum height be attained?\r<BR>\n" );
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document.write( "Answer:<BR>\n" );
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document.write( "Show work in this space.	<BR>\n" );
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document.write( "The ball spends half of its flight time going up and the other half of its flight time <BR>\n" );
document.write( "falling back to Earth.  So it rises for 2 seconds and falls for 2 seconds.<BR>\n" );
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document.write( "Since it rises for 2 seconds you can put 2 seconds into the height equation that we found for <BR>\n" );
document.write( "part a) and find the height at that time:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+-16t%5E2+%2B+64t+=+-16%2A%282%29%5E2+%2B+64%2A2+=+%28-16%2A4%29%2B%28128%29+=+64 ALIGN=MIDDLE  ALT=\"s+=+-16t%5E2+%2B+64t+=+-16%2A%282%29%5E2+%2B+64%2A2+=+%28-16%2A4%29%2B%28128%29+=+64\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The ball is at a height of 64 feet at the peak of its flight.<BR>\n" );
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document.write( "I hope this helps you to understand flight concepts.  The same basic idea applies to<BR>\n" );
document.write( "rocket launches, basketball shots, and football punts and passes.  </SPAN>\n" );
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