How Can Fermat's Principle and Snell's Law Help You Find a Car Key in Water?

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Fermat's Principle and Snell's Law can be applied to determine the distance of a car key submerged in water and the time it takes for light to reach the observer's eyes. The key's position can be calculated using the refractive index of water and the observer's height. The discussion highlights the importance of the actual light path, which minimizes travel time, illustrating Fermat's Principle. Additionally, the problem explores how to find the optimal distance for light reflection and its relation to Snell's Law. The thread concludes with a request for further assistance on specific parts of the problem.
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1. Your car key has fallen to the bottom of a clear pool of water (n = 1.33) with a uniform depth of 3.0 m. You are standing at the edge of the pool and your eyes are a height of 1.5 m above the surface. You can see the key by staring at an angle 60o below horizontal. (A) How far is the key from the edge of the pool? (B) i. How much time does it take light reflecting off the key to reach your eyes? ii. How much time would it take the light to travel to your eyes if it traveled in a straight line (instead of refracting at the surface)? iii. How much time would it take the light to travel if it exited the water vertically, then turned and traveled directly to your eyes? iv. What do your results suggest is special about the actual path the light ray follows? This is known as Fermat’s Principle. (C) Imagine a light ray that reflects off the key, strikes the surface of the water a distance x from the edge of this pool, and then bends to travel towards your eye. How much time would it take the light to travel to your eyes in this general case? (D) Use your result from part C to find the distance x which minimizes the time it takes for the light to reach your eyes. What does this tell you about Snell’s Law?



Homework Equations



Snell's Law


3. The Attempt at a Solution
I can easily solve part a and b using algebra and snells law but I'm wondering about how to approach part c and d. thank you

 
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