When recovering a position vector from a known velocity vector by integration
MATHEMATICS · Single correct · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
When recovering a position vector r(t) from a known velocity vector v(t) by integration, which of the following statements best justifies the appearance of an additional constant vector C in r(t)=∫v(t)dt+C?
(a) C represents the initial velocity carried through a second integration, because two integrations are required to go from acceleration to position.
(b) The constant vector C is added only when the particle does not start at the origin; if r(0)=0, no constant is needed.
(c) Each antiderivative of v(t) differs by a constant vector, so C accounts for all possible initial positions.
(d) Integrating each component of v(t) introduces a separate scalar constant, which are collected into a single vector C only when the component functions of v(t) are non-zero.