Velocity and Acceleration

Calculus 3 · Axiom Academy

LESSON Velocity and Acceleration Understanding motion through derivatives: from position to velocity to acceleration, and decomposing into tangential and normal components A position vector describes the location of a particle at time t . As t varies, the tip of the vector traces out a curve in space. The first derivative of position is velocity . The velocity vector is always tangent to the path and points in the direction of motion. Its length is the speed. Direction: Tangent to the curve (direction of motion) This is a crucial distinction: velocity is a vector (it has direction), while speed is a scalar (just a number). The second derivative of position (or the first derivative of velocity) is acceleration . It tells us how velocity is changing: hold the tails of and together, divide the gap by , and shrink — the quotient settles onto . Notice where ends up pointing: never outward, always into the bend. The velocity is turning that way, so the change in velocity must lean that way too. 4. Tangential and Normal Components We can decompose acceleration into two meaningful parts: one part that changes speed (tangential) and one part that changes direction (normal). They are perpendicular, so is the hypotenuse: . 5. Constant Speed: All Turn, No Push Take the sharpest case of that last remark — a particle going around a circle at constant speed. Since the speed never changes, , so every bit of the acceleration is normal. It is pure turning.

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