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Lifeguard's Fastest Path to a Swimmer: A Case of Fermat's Principle

Explore Fermat's Principle of Least Time and how it applies to a lifeguard's path on a beach to reach a swimmer in the least amount of time. Learn about wave refraction, Huygens' explanation, Snell's Law, and more.

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Lifeguard's Fastest Path to a Swimmer: A Case of Fermat's Principle

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  1. Question of the Day: Which path should a lifeguard (L) take at a beach if she wants to reach the swimmer (S) in the least time? Assume she can run faster in sand than she can swim in water. L sand water S

  2. This is an application of Fermat’s Principle of Least Time

  3. ___________________is the _____________of a wave as it passes_____________________________________. Refraction bending from one medium into another transmitted Ex: It is the ___________________ wave that is __________________ . refracted ray wavefront ____________ diagram: ____________ diagram: medium 1 surface medium 2 reflected Note: The _________________ wave is not shown here.

  4. Huygens Ex. Refraction explained using _______________ wavefronts.  Each point on the surface serves as a _____________ source for new ________________ in the second medium. point "wavelets"

  5. color When a light wave refracts, its ____________ does not change. This means its __________________ does not change. But the _______________ does change. frequency speed In medium 2: In medium 1: v = fl v = f l changes = (same) changes If speed increases, l ___________________ . If speed decreases, l ___________________ . increases decreases all waves This is true for ______________________ .

  6. Top view of wheels on an axle: = pavement = fast medium for wheels grass = slow medium for wheels This wheel slows down while other wheel is still moving fast

  7. Refraction of light causes mirages. mirage: sand acts like a mirror

  8. Water wave refraction:

  9. Refraction only occurs when a wave: 1/______________________________, AND 2/ enters the new medium _____________ (at an angle_________________________.) changes speed obliquely other than 00 incident ray reflected ray normal medium 1 boundary medium 2 refracted ray

  10. transmitted The _______________ ray is not refracted if EITHER there is: incident • _____________ • _____________ no speed change (n1 = n2) 1 2 transmitted …OR if: 2. ____________ ____________ q1 = 00 incident 1 not oblique 2 transmitted

  11. Important points: reflected The _____________ray will ___________ be in the quadrant that is _______ _______________of the boundary as the incident ray. always normal incident ray on the same side 1 boundary 2 The ______________ ray will ____________be in the quadrant that is __________ of the quadrant of the incident ray. refracted No rays always _________ in this quadrant! opposite

  12. Ex: A ray is shown incident on a boundary, shown by a darker line. Give the quadrant of the reflected and the refracted ray. C/ A/ II I refracted II I reflected III IV III IV reflected refracted B/ D/ reflected II I II I refracted III IV III IV reflected refracted

  13. Snell’s Law ________________tells exactly how much refraction occurs. angle of incidence q1 n1 n2 = n1sinq1 n2 sinq2 q2 medium 1 medium 2 angle of refraction Calculators must be in ______________mode, and not in ___________ mode. degrees radians

  14. Page 2: bottom Which medium refracts waves the most? diamond Which medium refracts waves the least? air

  15. Ex 1: Find the angle of refraction, q2 incident 600 air Lucite n1sinq1 = n2 sinq2 q2 1.5 sinq2 1.00 sin600 = 1.00 (0.866) = 1.5 sinq2 0.5774 = sinq2 sin-1(0.5774)= q2 350= q2

  16. Ex 2: Find the absolute index of refraction of the second medium, n2 300 zircon unknown 460 n1sinq1 = n2 sinq2 1.92 sin300 = n2 sin460 1.92 (0.500)= n2 (0.71934) 1.92 (0.500)= n2 0.71934 What medium is the unknown? 1.33 = n2

  17. Page 2: bottom

  18. > If v1 ____ v2 (or n1 ____ n2): < 1 Decreasing v:  _________________ angle  bends__________________ 2 decreasing towards normal initial direction If v1 ____ v2 (or n1 ___ n2): < > 1 • Increasing v: • __________________angle  bends___________________ 2 increasing away from normal initial direction

  19. 1 2 2 1 2 1 2 1 1 2 1 2 Ex: In which medium is light faster? G. D. A. H. E. B. I. F. C.

  20. apparent sun The atmosphere true sun is below horizon Earth your horizon The sun has already set when you see it setting!

  21. Ex. Pools of water apparent depth actual depth i o farther Objects in pools are _______________ than they appear!

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