Angle to Shoot Bow & Arrow at Object 1m Deep in Lake

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AI Thread Summary
A man is trying to determine the angle to shoot a bow and arrow at a treasure chest located 1 meter deep in a lake, while his eyes are 2 meters above ground and he looks at the chest at a 30-degree angle below the horizontal. The discussion highlights the use of the law of refraction, which involves calculating the angles of incidence and refraction with respect to a vertical normal. The user struggles with the calculations, particularly with the substitution into the refraction formula, leading to an impossible result of sin-1 greater than 1. There is a suggestion to clarify the diagram and properly label the refractive indices for air and water to assist in solving the problem. The conversation emphasizes the importance of incorporating the given heights into the calculations for accurate results.
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Homework Statement


A man whose eyes are 2 m above the ground is looking into a clear lake. He sees a treasure chest on the bottom of the lake, which is 1 m deep. His eyes make an angle of 30 degrees below the horizontal as he looks at the chest.

If he wants to shoot the chest with a bow and arrow (so he can pull it to the surface), at what angle should he shoot the arrow?

State the angle with respect to the horizontal.

Homework Equations


I learned the law of refraction was nisin(θi) = nrsin(θr). The angle is with respect to a VERTICAL normal. so \| the angle between those two lines is the angle of the ray going in, |\ that's the angle going out.

i means incidient, r means refracted.


The Attempt at a Solution


I can't get an answer for this, but I'm not sure what I'm doing wrong.

I'm assuming that when the guy is looking at the object, he's seeing the refracted object. The real object is actually in a different spot. So the angle that his eyes make with respect to the vertical should be θr.

If the guy's eyes make an angle of 30 degrees from the horizontal, then the angle with respect to a vertical normal must be 60 degrees.

Then using the equation given I should be able to solve for θi. But I can't, because it gives sin-1 > 1, which is impossible.

Also, I didn't use the heights given to me at all and I think I'm supposed to. What am I doing wrong?
 
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You must have mixed up the substitution into the refraction formula. On your diagram, label the air side 1 and write n1 = 1. On the water side 2, n2 = 1.33. Show your formula with numbers if you need more help!
 
Check out this amaturistic diagram, hope it helps?:confused:
 

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Please tell me is this anywhere close?
 
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It would be quicker if you uploaded it to a photo site like photobucket.com and then posted a link here. If you put IMG and /IMG (both in square brackets) around the link, the image will appear right in the post.
 

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