Question

A particle moves along a straight line and its position at time t is given by

s(t)=2t^3−24t^2+90t t>=0 where s is measured in feet and t in seconds.
(B) Use interval notation to indicate the time interval(s) when the particle is speeding up and slowing down.


(A) Use interval notation to indicate the time interval or union of time intervals when the particle is moving forward and backward

Answer 1

\[s(t)=2t^3−24t^2+90t\]
\[v(t)=s'(t)=6t^2−48t+90=6(t-3)(t-5)\]
\[a(t)=s''(t)=12t−48=12(t-4)\]

(A) forward 0<t<3 or t>5, backward 3<t<5
(B) speeding up t>4, slowing down 0<t<4