Part 5/8:
a_t = v' = \frac{d v}{dt}
]
- Normal acceleration ((a_n)): related to curvature and the square of the speed
[
a_n = \kappa v^2
]
Thus, the total acceleration vector can be expressed as:
[
\mathbf{a} = a_t \mathbf{T} + a_n \mathbf{N}
]
This decomposition allows for an intuitive understanding: the tangential component influences how fast the particle accelerates along the path, while the normal component causes it to change direction, effectively "turning" the particle.
Curvature and Its Relation to Motion
Curvature (\kappa) quantitatively measures how sharply a curve bends at a point. The formula uncovered in earlier sections relates curvature to the derivative of the unit tangent vector with respect to arc length:
[