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 1.5 m deep. You're standing on a small stool that places your eyes 3.9 m above the bottom of the pool. As you look at the target, your gaze is 35 degrees below horizontal. 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.

At what angle below horizontal should you throw the spear in order to hit the target?
Express your answer using two significant figures.
target  = ______________degrees

Explanation / Answer

The situation is as shown in figure The angle of glance is i = 35o The depth of water is d = 1.5 m The height of the observer above the water is h = 3.9 m - 1.5 m = 2.4 m Refractive index of water is n = 1.33 Apply Snell's law at the water-air interface sin(90-i) = nsin(90-r) cosi = ncosr cos35 = (1.33) cosr r = 52o From the laws of triangle D = h/tani     = 2.4 m / tan35     = 3.43 m Also D' = d/tanr      = 1.5 m / tan52      = 1.17 m Therefore the angle at which the spear to be thrown is = tan-1[(h+d)/(D+D')] = tan-1[3.9 m/4.6 m] = 40.29o From the laws of triangle D = h/tani     = 2.4 m / tan35     = 3.43 m Also D' = d/tanr      = 1.5 m / tan52      = 1.17 m Therefore the angle at which the spear to be thrown is = tan-1[(h+d)/(D+D')] = tan-1[3.9 m/4.6 m] = 40.29o

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