document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170979>Question 1170979</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A ball is thrown off a 30 m cliff at 20 m/s at an angle of 60 degrees above the horizontal<BR>\n" );
document.write( "What is the maximum height of the object? <BR>\n" );
document.write( "What is the total hang time of the ball?<BR>\n" );
document.write( "What is the final velocity of the ball?<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 #795866 by </B><A HREF=/tutors/aboutme.mpl?userid=Solver92311><B>Solver92311(821)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Solver92311><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=Solver92311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=795866\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Since you provided no information on temperature, precipitation, air density, wind speed, or wind direction, I have to presume that this experiment was conducted in some sort of huge evacuated container and the effects of the atmosphere on the projectile are not to be considered.\r<BR>\n" );
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document.write( "The height in meters of a projectile near the surface of the earth at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t\"> seconds after it is launched is given by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (1) \ \ \ \ \ \ \ \ h(t) =\ -4.9t^2\ +\ \sin(\theta)v_ot\ =\ h_o\">\r<BR>\n" );
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document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large v_o\"> is the initial velocity in meters per second of the projectile, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \theta\"> is the angle of launch, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large h_o\"> is the initial height in meters. Then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sin(\theta)v_o\"> is the vertical component of the initial velocity.\r<BR>\n" );
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document.write( "The instantaneous vertical velocity is given by the first derivative of the height function:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (2) \ \ \ \ \ \ \ \ \frac{dh}{dt}\ =\ v(t)\ =\ -9.8t\ +\ \sin(\theta)v_o\">\r<BR>\n" );
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document.write( "The maximum height of the projectile is the value of the height function at the time the velocity function is zero.  So set equation (2) equal to zero and solve for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t_{max}\"> and then calculate the value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large h\(t_{max}\)\">.\r<BR>\n" );
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document.write( "The projectile will be in the air for the amount of time it takes for the ball to reach zero height.  Set the height function equal to zero and then solve for the positive root which will be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t_{final}\">.\r<BR>\n" );
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document.write( "The final velocity will be the value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large v\(t_{final}\)\">. \r<BR>\n" );
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document.write( "<b>UPDATE</b>\r<BR>\n" );
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document.write( "The final vertical velocity will be the value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large v\(t_{final}\)\">.\r<BR>\n" );
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document.write( "The horizontal velocity is a constant <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \cos(\theta\)v_o\">\r<BR>\n" );
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document.write( "The final velocity magnitude is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sqrt{v\(t_{final}\)^2\,+\,\(\cos(\theta\)v_o\)^2}\">\r<BR>\n" );
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document.write( "The final velocity direction is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 180\ +\ \arctan\(\frac{v\(t_{final}\)}{\(\cos(\theta\)v_o}\)\"><BR>\n" );
document.write( "Make sure your calculator is set to degrees when you do the arctan calculation.\r<BR>\n" );
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document.write( "You can do your own arithmetic.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( "From <https://www.algebra.com/cgi-bin/upload-illustration.mpl> <BR>\n" );
document.write( "I > Ø<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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