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 4.0 m above the bottom of the pool. As you look at the target, your gaze is 30 below horizontal.

Part A

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. Index of refraction of water is 1.33.

Answer with 2 sig figs. Thank you, will rate!

Explanation / Answer

Angle of glance, i = 30 degree

Depth of water, d = 1.5 m

The height of the observer above the water is h = 4.0 m - 1.5 m = 2.5 m

Refractive index of water is n = 1.33

Apply Snell's law at the water-air interface

n1 * sin(90- i) = n2 * sin(90 - r)

1 * cos(i) = 1.33 * cos(r)

r = cos^-1(cos(30)/1.33)

r = 49.4

D = h/tan(i)

D = 2.5 / tan(30)

D = 4.33 m

Also

D' = d/tan(r)

D' = 1.5 m / tan(49.4)

D' = 1.29 m

Therefore the angle at which the spear to be thrown is

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

Theta = tan-1[4.0 m/5.62 m]

Theta = 35.44 degree

Angle below horizontal should you throw the spear in order to hit the target, theta = 35.44 degree

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