MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
For a certain particle, engineers record its velocity as ⟨6,4t⟩ with position measured in meters and time in seconds. They later decide to reparameterize time by setting s=t−3, so that s=0 corresponds to the moment t=3. The particle’s position at t=3 is (1,−5).
(a) Explain how the new parameter s alters the coordinate functions of the particle’s path while leaving the geometric curve unchanged.
(b) Determine the particle’s position when s=0 under the new parameterization and justify your answer using the given information.
(c) Discuss whether the particle’s speed depends on the choice between the parameters t and s. Provide an explanation supported by appropriate reasoning.