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 1.6 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 34° 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.

(a) At what angle below the horizontal should you throw the spear in order to hit the target?

(b) In a double or nothing play, the pool is partially covered by a sheet and the target repositioned under the sheet. A mirror is placed on the wall on the other side of the pool (2.0 m from where you stand). When you look at the target in the mirror your gaze is 15° below the horizontal. Do you take the double or nothing challenge, and if so at what angle should you throw it to win (you can assume that the spear will easily penetrate the sheet)?

Explanation / Answer

a) If you are gazing at 34 degrees under horizon, your gaze is meeting the surface of waer at 56 degrees By n1sin(theta)1 = n2sin(theta)2 If 1 is air, n1 is 1.000, (theta)1 is 56 degrees, n2 is 1.333 solve for (theta)2 theta2 = sin^-1(sin56 x 1.000 / 1.333) = 38.5 degrees height from eye to surface is 2.4 m, and from surface to bottom is 1.6 m Using trignometry, solve for horizontal distance in each region. For eye to surface, tan56 x 2.4 = 3.56 m For surface to bottom, tan38.5 x 1.6 = 1.27 m Add these two distances together 3.56 + 1.27 = 4.83 m target is 4.83 m in front and 4.00 down, thus tan^-1(4.00/4.83) = 39.6 degrees You must aim 39.6 degrees below horizon to hit the target. b) Do not take the challenge. You cannot calculate the position of target since you do not know how much mirror is tilted. You cannot assume it is vertical since if you are looking 15 degrees down, and mirror is 2 m in front of you, tan15 x 4 = 1.07 m You would be looking at only 1.07 m below where you are standing, not even close to bottom of the pool. There are not enough values provided for calculation of position, so do not take the challenge, take whatever you got already, and leave. Hope it helps :D

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