document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191828>Question 1191828</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A player kicks a football at an angle of 35° with the horizontal and with an initial velocity of 14.8 m/s. A second player standing at a distance of 26.0 m from the first player in the direction of the kick starts running to meet the ball at the instant it is kicked. <BR>\n" );
document.write( "(a) At what rate should the second player accelerate to catch the ball? <BR>\n" );
document.write( "(b) What would be the speed of the second player upon catching the ball? </SPAN>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "t = elapsed number of seconds\r<BR>\n" );
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document.write( "x = horizontal distance the ball is from the kicker<BR>\n" );
document.write( "y = vertical distance the ball is from the kicker<BR>\n" );
document.write( "distances are in meters\r<BR>\n" );
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document.write( "vx = horizontal speed<BR>\n" );
document.write( "vy = initial vertical speed<BR>\n" );
document.write( "Speeds are in m/s.\r<BR>\n" );
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document.write( "Draw a right triangle. Mark the angle of elevation to be 35 degrees. <BR>\n" );
document.write( "The hypotenuse of this triangle is the initial velocity 14.8 m/s.<BR>\n" );
document.write( "The horizontal leg and vertical leg are vx and vy respectively.\r<BR>\n" );
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document.write( "Using trig ratios, we can say:<BR>\n" );
document.write( "cos(theta) = adjacent/hypotenuse<BR>\n" );
document.write( "cos(35) = (horizontal speed)/(diagonal speed)<BR>\n" );
document.write( "cos(35) = (vx)/(14.8)<BR>\n" );
document.write( "vx = 14.8*cos(35)<BR>\n" );
document.write( "vx = 12.123450\r<BR>\n" );
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document.write( "and also,<BR>\n" );
document.write( "sin(theta) = opposite/hypotenuse<BR>\n" );
document.write( "sin(35) = (vertical speed)/(diagonal speed)<BR>\n" );
document.write( "sin(35) = (vy)/(14.8)<BR>\n" );
document.write( "vy = 14.8*sin(35)<BR>\n" );
document.write( "vy = 8.488931\r<BR>\n" );
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document.write( "We can use those speeds to get<BR>\n" );
document.write( "x = vx*t = 12.123450*t<BR>\n" );
document.write( "y = vy*t - 0.5*g*t^2 = 8.488931*t - 0.5*g*t^2<BR>\n" );
document.write( "where g = 9.81 m/s^2 approximately is the acceleration of gravity<BR>\n" );
document.write( "For more info, check out the projectile motion equations.\r<BR>\n" );
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document.write( "Let's plug in that approximate value of g<BR>\n" );
document.write( "y = 8.488931*t - 0.5*g*t^2<BR>\n" );
document.write( "y = 8.488931*t - 0.5*9.81*t^2<BR>\n" );
document.write( "y = 8.488931*t - 4.905*t^2<BR>\n" );
document.write( "y = -4.905*t^2 + 8.488931*t\r<BR>\n" );
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document.write( "Now let's find out when the ball will hit the ground.<BR>\n" );
document.write( "This occurs when y = 0.<BR>\n" );
document.write( "y = -4.905*t^2 + 8.488931*t<BR>\n" );
document.write( "-4.905*t^2 + 8.488931*t = y<BR>\n" );
document.write( "-4.905*t^2 + 8.488931*t = 0<BR>\n" );
document.write( "t(-4.905t + 8.488931) = 0<BR>\n" );
document.write( "t = 0 or -4.905t + 8.488931 = 0<BR>\n" );
document.write( "t = 0 or -4.905t = -8.488931<BR>\n" );
document.write( "t = 0 or t = -8.488931/(-4.905)<BR>\n" );
document.write( "t = 0 or t = 1.730669\r<BR>\n" );
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document.write( "We already know the ball is on the ground at the initial time t = 0 seconds (when it was kicked), so we'll ignore that solution.<BR>\n" );
document.write( "The only solution is roughly t = 1.730669 seconds.<BR>\n" );
document.write( "This is the amount of time the runner has to catch the ball.\r<BR>\n" );
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document.write( "Plug this t value into the x position equation found earlier.<BR>\n" );
document.write( "x = 12.123450*t<BR>\n" );
document.write( "x = 12.123450*1.730669<BR>\n" );
document.write( "x = 20.981679<BR>\n" );
document.write( "This is the horizontal distance the ball travels if we ignore wind resistance. <BR>\n" );
document.write( "Accounting for wind resistance will greatly complicate the problem.\r<BR>\n" );
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document.write( "The runner is 26 meters ahead of the kicker and needs to get to the 20.981679 marker. The runner needs to turn back toward the kicker to run 26-20.981679 = 5.018321 meters.\r<BR>\n" );
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document.write( "To catch the ball in time, they need to run about 5.018321 meters in roughly 1.730669 seconds.\r<BR>\n" );
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document.write( "I'm assuming the runner does not get a head start, which means the runner's initial velocity is vi = 0\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "a = acceleration<BR>\n" );
document.write( "a = (x - vi*t)/(0.5*t^2) ... variation of a kinematics equation<BR>\n" );
document.write( "a = (5.018321 - 0*1.730669)/(0.5*(1.730669)^2)<BR>\n" );
document.write( "a = 3.350892\r<BR>\n" );
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document.write( "Answer: <font color=red>Approximately 3.350892 m/s^2</font>\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "Use the values mentioned in the previous part.\r<BR>\n" );
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document.write( "velocity = acceleration*time ... formula works only if vi = 0<BR>\n" );
document.write( "velocity = 3.350892*t<BR>\n" );
document.write( "velocity = 3.350892*1.730669<BR>\n" );
document.write( "velocity = 5.799285\r<BR>\n" );
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document.write( "Answer: <font color=red>5.799285 m/s approximately</font><BR>\n" );
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