A transportation civil engineer is designing a track layout for a new high-speed passenger railway. The horizontal position profile of the track alignment transitions from a straight access track into a curved bypass, modeled by the piecewise function:
where and are measured in hundreds of meters.
- Analytically determine if the track layout is continuous at the junction point .
- Determine if the track layout is differentiable at by comparing the left-hand and right-hand derivative limits.
- Based on differentiability and lateral dynamics, explain why this transition geometry is either safe or hazardous for a passenger train traveling at high speeds.