I Wish The Car Stayed Still

Calculus Level 4

A ladybug is climbing on a Volkswagen Beetle (a type of car). In its starting position, the surface of the car is represented by the unit semicircle \(x^{2} + y^{2} = 1, y \geqslant 0 \) in the \(xy\)-plane. The \(x\)-axis is the road. At time \( t = 0 \) the ladybug starts at the front bumper \( (1, 0) \), and crawls counterclockwise on the surface of the car at a unit speed relative to the car. At the same time, the car moves to the right at speed 10.

If the speed of the bug at \( t = \dfrac{\pi}{4} \) can be represented as \( S \), find \( \left \lfloor 10S \right \rfloor \).


Credit: Problem information is adapted from MIT OCW.
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