document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=934070>Question 934070</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A toy rocket is launched from the ground level with an initial velocity of 96ft/s. After how many seconds will the rocket hit the ground?\r<BR>\n" );
document.write( "\n" );
document.write( "In this problem, I understand that I have to make a parabola. This means that the 96ft/s will be the peak of the parabola and the x axis should mark the start and end point (when it hits the ground). However, I don't know how to make the algebraic expression that goes with it. If I can make the algebraic expression, I know that I have to use (-1)b/2a to find the x intercept, and then plug that number in to find the other x intercept. \r<BR>\n" );
document.write( "\n" );
document.write( "Please help... </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #567203 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=567203\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To solve the problem, you need a little more knowledge of physics.<BR>\n" );
document.write( "On or near the surface of planet Earth,<BR>\n" );
document.write( "anything not being held/pushed up falls down<BR>\n" );
document.write( "(towards the center of the Earth) with an acceleration of about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32 ALIGN=MIDDLE  ALT=\"32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ft%2Fs%5E2 ALIGN=MIDDLE  ALT=\"ft%2Fs%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Toy rockets get a very short push, to get an initial speed,<BR>\n" );
document.write( "and then they are not being pushed up any more.<BR>\n" );
document.write( "The upwards velocity of the rocket will decrease by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32 ALIGN=MIDDLE  ALT=\"32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ft%2Fs ALIGN=MIDDLE  ALT=\"ft%2Fs\"   DISABLED_style= \"max-width:90%; height:auto;\" > every second.<BR>\n" );
document.write( "The graph of upwards velocity as a function of time looks like this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-1%2C9%2C-100%2C100%2C-32x%2B96%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-1%2C9%2C-100%2C100%2C-32x%2B96%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The upwards velocity starts a s 96 ft/s at time zero,<BR>\n" );
document.write( "is zero at 3 seconds, and is negative (rocket falling down) after 3 seconds.<BR>\n" );
document.write( "It takes 3 seconds <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=96%2F32 ALIGN=MIDDLE  ALT=\"96%2F32\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the rocket to get to the highest point,<BR>\n" );
document.write( "and another 3 seconds for it to return to the height where it started<BR>\n" );
document.write( "(disregarding air resistance, of course),<BR>\n" );
document.write( "so the total flight time is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%286%29 ALIGN=MIDDLE  ALT=\"highlight%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > seconds.<BR>\n" );
document.write( " <BR>\n" );
document.write( "A math teacher will have told you that on Earth<BR>\n" );
document.write( "the height (in feet) as function of time since launch (in seconds)<BR>\n" );
document.write( "of a projectile launched upwards with an initial speed <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;\" > ft/s<BR>\n" );
document.write( "from an initial height <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;\" > feet is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=h%5B0%5D%2Bv%5B0%5Dt-16t%5E2 ALIGN=MIDDLE  ALT=\"h%28t%29=h%5B0%5D%2Bv%5B0%5Dt-16t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The teacher will have told you that the graph of that equation is a parabola,<BR>\n" );
document.write( "and wouldn't you like to believe and memorize that equation?<BR>\n" );
document.write( "Using that equation, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5B0%5D=0 ALIGN=MIDDLE  ALT=\"h%5B0%5D=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5B0%5D=96 ALIGN=MIDDLE  ALT=\"v%5B0%5D=96\"   DISABLED_style= \"max-width:90%; height:auto;\" > you get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=96t-16t%5E2 ALIGN=MIDDLE  ALT=\"h%28t%29=96t-16t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and the rocket is on the ground when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=0 ALIGN=MIDDLE  ALT=\"h%28t%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "which will happen for the solutions to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=96t-16t%5E2=0 ALIGN=MIDDLE  ALT=\"96t-16t%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16t%286-t%29=0 ALIGN=MIDDLE  ALT=\"16t%286-t%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28t=0%2C%22or%22%2Ct=highlight%286%29%29 ALIGN=MIDDLE  ALT=\"system%28t=0%2C%22or%22%2Ct=highlight%286%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "A physics teacher may tell you that you need to know calculus to derive that equation.<BR>\n" );
document.write( "I would tell you that it comes very naturally from the graph.<BR>\n" );
document.write( "Since upwards velocity multiplied times seconds in flight equals gain in height,<BR>\n" );
document.write( "the gain in height is the area above the x-axis<BR>\n" );
document.write( "and below the graph of upwards velocity as a function of time.<BR>\n" );
document.write( "For a real rocket that could maintain the constant 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;\" > ,<BR>\n" );
document.write( "the graph would be a flat line, and<BR>\n" );
document.write( "the gain in height as a function of time would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29-h%5B0%5D=v%5B0%5D%2At ALIGN=MIDDLE  ALT=\"h%28t%29-h%5B0%5D=v%5B0%5D%2At\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the area of a rectangle under the flat line:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C200%2C-.1%2C0.8%2C-.1%2C0.5%2C%0D%0Agrid%280%29%2Cline%280%2C0.35%2C0.8%2C0.35%29%2C%0D%0Agreen%28line%280.6%2C0%2C0.6%2C0.35%29%29%2C%0D%0Alocate%280.58%2C0%2Cgreen%28t%29%29%2C%0D%0Alocate%280.2%2C0.25%2Cv%5B0%5D%2At%29%2C%0D%0Alocate%28-0.05%2C0.4%2Cv%5B0%5D%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In the case of the toy rocket, the graph is a straight line sloping down,<BR>\n" );
document.write( "so you have to subtract a right triangle from that rectangle.<BR>\n" );
document.write( "The triangle to be subtracted has legs measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32t ALIGN=MIDDLE  ALT=\"32t\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so its area is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2At%2A%2832t%29=16t%5E2 ALIGN=MIDDLE  ALT=\"%281%2F2%29%2At%2A%2832t%29=16t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C200%2C-.1%2C0.8%2C-.1%2C0.5%2C%0D%0Agrid%280%29%2Cline%280%2C0.35%2C0.8%2C0.35%29%2C%0D%0Agreen%28line%280.6%2C0%2C0.6%2C0.35%29%29%2C%0D%0Alocate%280.58%2C0%2Cgreen%28t%29%29%2C%0D%0Alocate%280.2%2C0.15%2Cv%5B0%5D%2At-16t%5E2%29%2C%0D%0Alocate%28-0.05%2C0.4%2Cv%5B0%5D%29%2C%0D%0Ared%28line%280%2C0.35%2C0.8%2C0.05%29%29%2C%0D%0Alocate%280.61%2C0.25%2C16t%29%2Carrow%280.63%2C0.25%2C0.63%2C0.35%29%2C%0D%0Aarrow%280.63%2C0.2%2C0.63%2C0.125%29%2C%0D%0Alocate%280.33%2C0.33%2C16t%5E2%29%2Clocate%280.31%2C0.4%2Ct%29%2C%0D%0Aarrow%280.3%2C0.38%2C0%2C0.38%29%2Carrow%280.34%2C0.38%2C0.6%2C0.38%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > so the change in height is the area of the trapezoid with green and red sides:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29-h%5B0%5D=+v%5B0%5D%2At-16t%5E2 ALIGN=MIDDLE  ALT=\"h%28t%29-h%5B0%5D=+v%5B0%5D%2At-16t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );