document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=557922>Question 557922</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Where does the -16 come from in the vertical motion formula? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #362851 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=362851\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Good question. Start with the general equation for vertical motion:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x = -\frac{1}{2}at^2 + v_0t + x_0\"> where a is the acceleration due to gravity.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This equation can be derived using integral calculus (I'll prove it next). The acceleration due to gravity can be derived from the equation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE F = m_1a = G\frac{m_1m_2}{R^2}\"> where G is the gravitation constant, m1 and m2 are the masses of some object and Earth respectively, and R is the radius of the Earth. Plugging in, we find that a is approximately -9.8 m/s^2 (about -32 ft/s^2). Half of that is -16 ft/s^2, which is where the -16 comes from.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To derive the vertical motion formula, start with the fact that\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE v = \int a\, dt = at + v_0\"> (the v0 is our constant of integration)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Integrating again,\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x = \int v\, dt = \int at + v_0\, dt = \frac{1}{2}at^2 + v_0t + x_0\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "And we are done.\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );