One of the contests at the school carnival is to throw a spear at an underwater

Question

One of the contests at the school carnival is to throw a spear at an underwater target lying flat on the bottom of a pool. The water is 1.00m deep. You're standing on a small stool that places your eyes 3.10m above the bottom of the pool. As you look at the target, your gaze is 30? (degrees) below horizontal. At what angle below horizontal should you throw the spear in order to hit the target?

Your raised arm brings the spear point to the level of your eyes as you throw it, and over this short distance you can assume that the spear travels in a straight line rather than a parabolic trajectory. Express your answer with the appropriate units.

Explanation / Answer

From snells law , n sin tetha = sin ( pi / 2 - alpha)

sin tetha = cos alpha / n

tan tetha = cos alpha / sqrt ( n^2 - cosalpha ^2)

For small triangle tan tetha = x / d

combining x = d tan tetha = d cos alpha / sqrt ( n^2 - cos alpha ^2)

from large triangle tan beta = h + d / ( x +(h / tan alpha))

substituing for x ; tan beta = (h + d ) / ( dcos alpha / sqrt ( n^2 - cos alpha^2) + h / tan alpha)

bysubstituing n = 1.33 , h = 3.10 , d = 1 , alpha = 30 degrees

tan beta = ( 3.10 +1) tan 30 (sqrt( 1.33^2 - cos 30^2) ) / ( sin 30 + 3.10sqrt ( 1.33^2 - cos30^2))

tan beta = 4.10* 0.577 *1.004 / (0.5 + 3.1124)

tan beta = 0.6574

beta = 33.320

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