Plz Help!  s(t)=t^3−8t gives the position in miles of a freight train where east is the positive direction and t is measured in hours?

Given s(t) you can find the velocity v(t) [how?].

a) When s(t) = 0, then t = [left for you. There are TWO answers. az made a mistake: t=2 is NOT a possibility]

The direction is given by the sigh of v(t). So at each of the times found above, what are the signs of v(t) ?

b) given v(t), whats a(t) ? When a(t) = 0, t = ? What is the sign of v at this time?

c) When |v(t)| decreases... think of its graph if need be.

Don't forget to vote me best answer for being the first to correctly walk you through w/o spoiling the answers! 

a.  v(t) = 3t^2 - 8.

The s(t) will be 0 at t = 0, at t = 2, and it would also be 0 at t = -2, but perhaps that doesn't come into the problem. 

Anyway, when t = 0 and s = 0, the train is moving westward.

When s = 0 and t = 2, the v(t) = 12 - 8 = +4, so the train is moving eastward.

b.  a(t) = 6t.

This is 0 only when t = 0.  The train is traveling westward, but its westward motion is about to slow down as t and a(t) turn positive.

c.  The train's westward motion slows down from t=0 until t = sqrt(8/3), at which time the westward motion stops and an eastward motion begins.  The eastward motion then speeds up forever, apparently.