Blending of Doppler's Effect and Calculus

Consider a standard Cartesian coordinate system (the xyxy plane). A sound source is moving on a parabolic path y2=4x\displaystyle{{ y }^{ 2 }=4x} with constant speed vs=v2\displaystyle{{ v }_{ s }=\cfrac { v }{ 2 } }. Here vv is velocity of sound in still air.

At time t=0t=0 the source is at the origin, and an observer is standing at rest at (1,0)\displaystyle{(-1,0)}. Find the time at which the observer hears the lowest frequency he'll hear from the source. The time TT can be expressed as T=ab+cln(d+e)v\displaystyle{T=\cfrac { a\sqrt { b } +c\ln { (\sqrt { d } +e) } }{ v } }

Find a+b+c+d+ea+b+c+d+e .

Details

  • Here a,b,c,d,ea,b,c,d,e are positive integers , and b,db,d are square free integer .
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