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Math Help - minimum time

  1. #1
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    minimum time

    show that the light waves traveling from P to Q follow the path of minimum time. i know i will be using velocity im jus not sure how, well attached is as far as i have gotten with this problem...i mean would i jus figure out the distance from P to Q and plug that into the formula for velocity, right?IMG.pdf

    thanks in advance...
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  2. #2
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    Quote Originally Posted by slapmaxwell1 View Post
    show that the light waves traveling from P to Q follow the path of minimum time. i know i will be using velocity im jus not sure how, well attached is as far as i have gotten with this problem...i mean would i jus figure out the distance from P to Q and plug that into the formula for velocity, right?IMG.pdf

    thanks in advance...
    t1 = sqrt(d1^2 + x^2)/v1
    t2 = sqrt[d2^2 + (a-x)^2]/v2
    Total timr t is
    t = t1 + t2 = sqrt(d1^2 + x^2)+ sqrt[d2^2 + (a-x)^2]
    For least time t
    dt/dx = x/v1[sqrt(d1^2 + x^2) - (a-x)/v2sqrt[d2^2 + (a-x)^2] = 0
    Tat leads to
    sinθ1/v1 = sinθ2/v2, which is the Snell's law.
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  3. #3
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    h1= d_1 cos(\theta_1) and h2= d_2 cos(\theta_2) so the total distance from P to Q is h1+ h2= d_1 cos(\theta_1)+ d_2 cos(\theta_2).
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