Tangential and Normal Components

Calculus 3 · Axiom Academy

LESSON Tangential and Normal Components Acceleration has two jobs at once — one part changes how fast you go, the other changes where you are pointed. 1. The Fundamental Decomposition For a particle moving along a curve with position vector , the acceleration vector can be decomposed into two perpendicular components — one along the direction of travel, one across it. is the unit tangent vector, the unit normal Points along the path. Changes speed — positive speeds the particle up, negative slows it down. Points toward the center of curvature — the concave side. Changes direction , never speed. The tangential and normal components can be calculated using projections and properties of the velocity and acceleration vectors. Drop a perpendicular from the tip of onto the line of : the piece along is a_T , and the leftover piece is a_N . the rate of change of speed = the projection of onto curvature × speed² = the perpendicular leftover is literally the rate at which the speed itself is changing. It carries a sign. is the shadow of on the direction of travel. : a sharper turn (bigger ) or a faster pass costs more normal acceleration. is everything left once the tangential piece is removed. It is never negative. 3. Special Case: Uniform Circular Motion

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