A particle moves along a coordinate line for seconds

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

A particle moves along a coordinate line for

0\le t\le3

seconds. Its acceleration is given by

a(t)=6t-2\;\text{(m/s}^2).

A candidate velocity function is proposed:

v(t)=3t^{2}-2t+C_1,

and a candidate position function is proposed:

s(t)=t^{3}-t^{2}+C_1 t+C_2,

where

C_1

and

C_2

are constants. (a) Show that

v'(t)=a(t)

for all

t

in the interval. (b) Given the initial condition

v(0)=4\text{ m/s}

, determine

C_1

. (c) Using the value of

C_1

from part (b), verify that

s''(t)=a(t)

for all

t

. (d) If the particle’s position at time

t=0

s(0)=1\text{ m}

, determine

C_2

and write the complete expressions for both

v(t)

and

s(t)

that satisfy all given information.