document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209422>Question 1209422</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The parametric equations<BR>\n" );
document.write( "x = vt \cos \theta,<BR>\n" );
document.write( "y = vt \sin \theta - \frac{1}{2} gt^2,<BR>\n" );
document.write( "describe the path of a projectile fired at an angle of theta, where v is the initial velocity, g denotes acceleration due to gravity, and t denotes time.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The path itself is a parabolic arch, and the distance the projectile reaches varies with the angle theta.   Find the maximum distance. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848804 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848804\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Find the time of flight:**\r<BR>\n" );
document.write( "\n" );
document.write( "* The projectile hits the ground when y = 0. <BR>\n" );
document.write( "* Set the y equation to zero: <BR>\n" );
document.write( "   vt * sin(theta) - (1/2) * g * t^2 = 0<BR>\n" );
document.write( "* Solve for t:<BR>\n" );
document.write( "   t(vt * sin(theta) - (1/2) * g * t) = 0<BR>\n" );
document.write( "   t = 0 (initial point) or t = (2 * v * sin(theta)) / g \r<BR>\n" );
document.write( "\n" );
document.write( "**2. Find the horizontal distance (range):**\r<BR>\n" );
document.write( "\n" );
document.write( "* Substitute the time of flight (t = (2 * v * sin(theta)) / g) into the x equation:<BR>\n" );
document.write( "   x = v * cos(theta) * (2 * v * sin(theta)) / g<BR>\n" );
document.write( "   x = (v^2 * 2 * sin(theta) * cos(theta)) / g\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Use the trigonometric identity:**\r<BR>\n" );
document.write( "\n" );
document.write( "* Recall the double-angle formula for sine: <BR>\n" );
document.write( "   sin(2 * theta) = 2 * sin(theta) * cos(theta)\r<BR>\n" );
document.write( "\n" );
document.write( "* Substitute into the range equation:<BR>\n" );
document.write( "   x = (v^2 * sin(2 * theta)) / g\r<BR>\n" );
document.write( "\n" );
document.write( "**4. Find the maximum distance:**\r<BR>\n" );
document.write( "\n" );
document.write( "* The maximum distance occurs when sin(2 * theta) is at its maximum value, which is 1.<BR>\n" );
document.write( "* This happens when 2 * theta = 90 degrees, or theta = 45 degrees.\r<BR>\n" );
document.write( "\n" );
document.write( "**5. Calculate the maximum distance:**\r<BR>\n" );
document.write( "\n" );
document.write( "* Substitute theta = 45 degrees into the range equation:<BR>\n" );
document.write( "   x_max = (v^2 * sin(90)) / g<BR>\n" );
document.write( "   x_max = (v^2) / g\r<BR>\n" );
document.write( "\n" );
document.write( "**Therefore, the maximum distance (range) of the projectile is achieved when the launch angle is 45 degrees, and the maximum distance is (v^2) / g.**<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );