# Blending of Doppler's Effect and Calculus

**Classical Mechanics**Level 4

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 .