# Blending of Doppler's Effect and Calculus

Classical Mechanics Level 5

Consider an Standard Cartesian Coordinate system (X-Y plane). If An 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 the position whose coordinates are $$\displaystyle{(-1,0)}$$ . Then Find the time interval in which observer Hears the minimum frequency of sound. If this time interval can be expressed as : $\displaystyle{T=\cfrac { a\sqrt { b } +c\ln { (\sqrt { d } +e) } }{ v } }$

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

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

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