document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1175536>Question 1175536</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A ball is thrown a projectile motion. It is known that the horizontal distance (range) the ball can travel is given by R = v^2/g sin 2x, where r is the range (in feet), v^0 is the initial speed (in/ft/s), x is the angle of elevation the ball is thrown, and g=32ft/s^2 is the acceleration due to gravity.\r<BR>\n" );
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document.write( "a. Express the new range in terms of the original range when an angle x (0<x<45 degrees) is doubled.\r<BR>\n" );
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document.write( "b. If a ball travels a horizontal distance of 20 ft when kicked at an angle of x with initial speed 20 √2ft/s, find the horizontal distance it can travel when you double x.<BR>\n" );
document.write( "Hint: use result of item (a) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850612 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=850612\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's tackle this projectile motion problem.\r<BR>\n" );
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document.write( "**a) Expressing the New Range in Terms of the Original Range**\r<BR>\n" );
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document.write( "* **Original Range:** R = (v^2 / g) * sin(2x)<BR>\n" );
document.write( "* **New Angle:** 90° - x\r<BR>\n" );
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document.write( "Let's find the new range, R', with the new angle:\r<BR>\n" );
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document.write( "* **New Range:** R' = (v^2 / g) * sin(2(90° - x))<BR>\n" );
document.write( "* R' = (v^2 / g) * sin(180° - 2x)\r<BR>\n" );
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document.write( "Now, we use the trigonometric identity: sin(180° - θ) = sin(θ)\r<BR>\n" );
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document.write( "* R' = (v^2 / g) * sin(2x)\r<BR>\n" );
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document.write( "Comparing this to the original range, R:\r<BR>\n" );
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document.write( "* R' = R\r<BR>\n" );
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document.write( "**Therefore, the new range is the same as the original range when the angle is changed from x to 90° - x.**\r<BR>\n" );
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document.write( "**b) Finding the Horizontal Distance When Doubling x**\r<BR>\n" );
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document.write( "1.  **Find the original angle x:**<BR>\n" );
document.write( "    * We're given: R = 20 ft, v = 20√2 ft/s, g = 32 ft/s²<BR>\n" );
document.write( "    * R = (v² / g) * sin(2x)<BR>\n" );
document.write( "    * 20 = ((20√2)² / 32) * sin(2x)<BR>\n" );
document.write( "    * 20 = (800 / 32) * sin(2x)<BR>\n" );
document.write( "    * 20 = 25 * sin(2x)<BR>\n" );
document.write( "    * sin(2x) = 20 / 25 = 4 / 5 = 0.8<BR>\n" );
document.write( "    * 2x = arcsin(0.8)<BR>\n" );
document.write( "    * 2x ≈ 53.13°<BR>\n" );
document.write( "    * x ≈ 26.565°\r<BR>\n" );
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document.write( "2.  **Find the new angle 2x:**<BR>\n" );
document.write( "    * The new angle is 2x, which we already found to be approximately 53.13°.\r<BR>\n" );
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document.write( "3.  **Find the new range R'':**<BR>\n" );
document.write( "    * R'' = (v² / g) * sin(2(2x))<BR>\n" );
document.write( "    * R'' = (v² / g) * sin(4x)<BR>\n" );
document.write( "    * We know 2x ≈ 53.13°, so 4x ≈ 106.26°<BR>\n" );
document.write( "    * R'' = ((20√2)² / 32) * sin(106.26°)<BR>\n" );
document.write( "    * R'' = 25 * sin(106.26°)<BR>\n" );
document.write( "    * R'' ≈ 25 * 0.96<BR>\n" );
document.write( "    * R'' ≈ 24 ft\r<BR>\n" );
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document.write( "**Therefore, the horizontal distance the ball can travel when you double x is approximately 24 feet.**<BR>\n" );
document.write( " </SPAN>\n" );
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