Question

A submarine is 2.92 ✕ 10² m horizontally from shore and 1.00 ✕ 10² m beneath the surface of the water. A laser beam is sent from the submarine so that the beam strikes the surface of the water 2.14 ✕ 10² m from the shore. A building stands on the shore, and the laser beam hits a target at the top of the building. The goal is to find the height of the target above sea level. (The index of refraction of water is n = 1.333.)

(a) Draw a diagram of the situation, identifying the two triangles that are important to finding the solution.

(b) Find the angle of incidence of the beam striking the water–air interface. (Give your answer to at least one decimal place.)
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(c) Find the angle of refraction. (Give your answer to at least one decimal place.)
____ °

(d) What angle does the refracted beam make with respect to the horizontal? (Give your answer to at least one decimal place.)
____ °

(e) Find the height of the target above sea level.
____ m

Answer 1

(a) (b) From the triangle under water, the angle of incidence theta_1 at the water surface tan theta_1 = 78 m/100 m Rightarrow theta_1 = 37.95 degree (c) Snell's law gives the angle of refraction theta_2 = sin^- 1 (n_water sin theta_1/ n_air) = sin^- 1 ((1.33) sin 37.95 degree/1) = 54.8 degree (d) The refracted beam makes angle phi = 90 - theta_2 = 90 - 54.8 degree = 35.1 degree with horizontal (e) In the triangle above water h = (214) tan phi = (214 m)tan 35.1 degree = 150.48 m