document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1038562>Question 1038562</A>: <I><SPAN STYLE=\"word-break: break-all;\"> When we kick a ball ,is it sure that ball always makes parabola if there is no air resistance?If yes,how can we find focus?<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 #653278 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=653278\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\"><BR>\n" );
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document.write( "You need the magnitude and direction of the initial velocity vector.  In the figure above, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \theta\"> represents the direction and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left\|\vec{v_o}\right\|\"> represents the magnitude of the initial velocity vector <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \vec{v_o}\">.  I'm going to assume that you are using the mks system.\r<BR>\n" );
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document.write( "The first thing that you need is to resolve the initial velocity vector into its horizontal and vertical components to get the magnitudes of the initial velocity in the horizontal direction and in the vertical direction.  Use the following formulae where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left\|\vec{v_y_o}\right\|\"> is the magnitude of the vertical component and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left\|\vec{v_x_o}\right\|\"> is the magnitude of the horizontal component.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left\|\vec{v_y_o}\right\|\ =\ \left\|\vec{v_o}\right\|\cdot\sin\theta\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left\|\vec{v_x_o}\right\|\ =\ \left\|\vec{v_o}\right\|\cdot\cos\theta\">\r<BR>\n" );
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document.write( "Since you are ignoring air resistance, we can presume that once the impulse of the kick has been applied, the only external force that acts upon the ball is gravity.  Gravity only acts in the vertical, so it is the vertical velocity component that becomes a part of the altitude function.  Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t\"> represent the time in seconds since the ball was kicked.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a(t)\ =\ -4.9t^2\ +\ \left\|\vec{v_y_o}\right\|t\ +\ a_o\">\r<BR>\n" );
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document.write( "If we also assume that the ball was resting on the ground when it was kicked, the initial altitude component, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a_o\">, is zero and can be ignored.  Thus:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a(t)\ =\ -4.9t^2\ +\ \left\|\vec{v_y_o}\right\|t\">\r<BR>\n" );
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document.write( "The graph of the altitude function is a parabola, but it is not the parabola that you seek.  The above is altitude as a function of time, whereas the function that you seek is altitude as a function of horizontal distance.  However, in order to determine the horizontal distances, we need to know the time it takes the ball to reach its maximum height.  This time is the value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t\"> at the vertex of the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a(t)\"> parabola. Recall that the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> value of the vertex point of a parabola with equation <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large ax^2\ +\ bx\ +\ c\"> is given by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x_v\ =\ \frac{-b}{2a}\">.  Hence:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ t_{max}\ =\ \frac{-\left\|\vec{v_y_o}\right\|}{-9.8}\">\r<BR>\n" );
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document.write( "The maximum height of the ball is then the value of the altitude function at time <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t_{max}\">, in other words:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a(t_{max})\ =\ -4.9t_{max}^2\ +\ \left\|\vec{v_y_o}\right\|t_{max}\">\r<BR>\n" );
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document.write( "The horizontal distance travelled by the ball at the time that the ball reaches the vertex of the parabola is simply [tex \Large t_{max}] multiplied by the horizontal component of the initial velocity, to wit:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ h(t_{max})\ =\ \left\|\vec{v_x_o}\right\|t_{max}\">\r<BR>\n" );
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document.write( "Now we have the coordinates of the vertex of our desired parabola, namely <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(h(t_{max}),a(t_{max})\right)\">.  Since we assume the ball was on the ground at a horizontal distance of zero when the ball was kicked, we also know that the point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (0,0)\"> is on our parabola.\r<BR>\n" );
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document.write( "Let us recall the equation of a parabola with a vertical axis:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ a(x\ -\ h)^2\ +\ k\">\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (h,k)\"> is the vertex of the parabola and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\">, the distance from the vertex to the focus is related to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\ =\ \frac{1}{4a}\">\r<BR>\n" );
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document.write( "Let's see what all this means given that the point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (0,0)\"> is on our parabola.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 0\ =\ a(0\ -\ h)^2\ +\ k\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ ah^2\ =\ -k\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a\ =\ \frac{-k}{h^2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ p\ =\ -\frac{h^2}{4k}\">\r<BR>\n" );
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document.write( "Therefore, the focus is at:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(h(t_{max}),a(t_{max})\ -\ \frac{\[h(t_{max})\]^2}{4\cdot a(t_{max})}\right)\">\r<BR>\n" );
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document.write( "Now all you need to do is the arithmetic.  Good luck.\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( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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