The velocity (in meters per second) of a particle moving along a straight line for…

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

The velocity

v(t)

(in meters per second) of a particle moving along a straight line for

0\le t\le 8

seconds is modeled by the piecewise function

v(t)=\begin{cases} t^2-6t+8, & 0\le t\le 3,\\ a\sin(t-3)+b, & 3<t\le 8, \end{cases}

where

a

and

b

are constants. (a) Given that

v

is continuous on

[0,8]

and that

v(3)=-1

, write an equation relating

a

and

b

. (b) If in addition

v(8)=2

, determine the numerical values of

a

and

b

. (c) With your values of

a

and

b

, show that there exists a time

t

(0,8)

at which

v(t)=0

. Provide a clear justification. (d) Interpret the result from part (c) in the context of the motion of the particle.