Two students analyze the motion of a particle whose velocity at time seconds is the vector

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

Two students analyze the motion of a particle whose velocity at time

t

seconds is the vector

\mathbf v(t)

. • Student A writes a position function

\mathbf r_A(t)=\langle F(t),G(t),H(t)\rangle

where

F'(t)=v_x(t)

G'(t)=v_y(t)

, and

H'(t)=v_z(t)

. • Student B writes

\mathbf r_B(t)=\langle F(t)+C_1,\,G(t)+C_2,\,H(t)+C_3\rangle

, where

C_1,C_2,

and

C_3

are real constants. Answer the following questions about the two proposed position functions. (a) Describe the mathematical relationship between

\mathbf r_A(t)

and

\mathbf r_B(t)

. Express your answer as simply as possible in vector form. (b) Explain why knowing the particle’s position

\mathbf r(t_0)

at a single instant

t_0

guarantees a unique position function, and discuss what would happen if no such information were provided. (c) Let

\mathbf v(t)=\langle 2t,\,\cos t,\,0\rangle

. Find two different position functions whose derivatives equal

\mathbf v(t)

and show explicitly how they differ by a constant vector.