A particle in the plane has velocity for all real

MATHEMATICS · Single correct · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

A particle in the plane has velocity v(t)=⟨2t, t2⟩\mathbf v(t)=\langle 2t,\,t^{2}\rangle for all real tt. Its acceleration is a(t)=⟨2, 2t⟩\mathbf a(t)=\langle 2,\,2t\rangle. For t>0t>0, which statement best describes the behavior of the particle’s speed? (a) The speed decreases for all t>0t>0 because v(t)⋅a(t)<0\mathbf v(t)\cdot \mathbf a(t)<0. (b) The speed increases on (0,1)(0,1) and decreases for t>1t>1 because v(t)⋅a(t)\mathbf v(t)\cdot \mathbf a(t) changes sign at t=1t=1. (c) The speed increases for all t>0t>0 because v(t)⋅a(t)>0\mathbf v(t)\cdot \mathbf a(t)>0. (d) The speed remains constant for all t>0t>0 because ∣v(t)∣|\mathbf v(t)| is independent of tt.

Related theory

Browse the problem library