# Blending of Doppler's Effect and Calculus

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

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

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

Details

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