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 1.10 m deep. You're standing on a small stool that places your eyes 3.20 m above the bottom of the pool. As you look at the target, your gaze is 30 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

Height of the observer h = 3.20 - 1.10 = 2.1 m

From snells law, n1 sin (90 - i) = n2 sin (90 - r)

1 * sin (90 - 30) = 1.33sin (90 - r)

r = cos-1(cos 30 / 1.33) = 49 deg

D = h / tan theta = 2.1 / tan 30 = 3.64 m

D' = d / tan r = 1.1 / tan 49 = 0.956 m

Angle at which spear is thrown

theta = tan-1 [h + d / (D + D')]

= tan-1 [2.1 + 1.1 / (3.64 + 0.956)]

theta = 34.8 deg

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