document.write( "Question 1188601: at what angle of firing will a projectile's range be equal to its maximum height? \n" ); document.write( "

Algebra.Com's Answer #819762 by ikleyn(53763) $\"\"$ $\"About$

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\n" ); document.write( "at what angle of firing will a projectile's range be equal to its maximum height?
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document.write( "So, we consider the staring point and the ending point of the projectile at the same horizontal ground level.\r\n" );
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document.write( "Let \"a\" be the angle of firing (with the horizon) and let \"v\" be the magnitude of the initial speed.\r\n" );
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document.write( "Then v*sin(a) is the initial vertical speed and v*cos(a) is (unchangeable, constant) horizontal speed.\r\n" );
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document.write( "Vertical speed is the linear function of time from the ground level to the highest point,\r\n" );
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document.write( "THEREFORE, the average vertical speed during the flight up is  .\r\n" );
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document.write( "The time moving up from the ground to the highest point is  t = ,\r\n" );
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document.write( "THERFORE, the maximum height is the product of the average vertical speed  \r\n" );
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document.write( "by the time to get the maximum height  \r\n" );
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document.write( "     = .\r\n" );
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document.write( "The range is the maximum horizontal distance  r =  = \r\n" );
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document.write( "Notice that the time moving horizontally is twice the time to get the highest point ! (which is obvious . . . )\r\n" );
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document.write( "Our equation is the equality of  and r,  or   = r,  which is\r\n" );
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document.write( "     = .\r\n" );
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document.write( "Canceling all common factors in the numerators and denominators, we get\r\n" );
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document.write( "     = 2*cos(a).\r\n" );
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document.write( "It implies  tan(a) = 4;  hence,  a = arctan(4) = 1.326 radians = 75.964 degrees.\r\n" );
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document.write( "ANSWER.  Under the imposed condition, the firing angle with the horizon is a = arctan(4) = 1.326 radians = 75.964 degrees.\r\n" );
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