Math trouble! Please help!?

Question

The position function is s(t)=t^3 - 3t^2 + 4 for an object moving along a linear path. distance = meters
Answer the following:
(a) On what interval(s) is the object moving to the right? to the left? changing direction?
(b) Determine the position, velocity, and acceleration when t=0 and t=3 seconds and then describe the motion at these points in time.
(c) Determine the distance traveled after the first 5 seconds.
(d) Determine the displacement after the first 5 seconds.

I really don't understand, so could you please explain how to solve the problem as well? Thank-you!

Answer 1

I assume that the lienaer path is left to right, with positive s to the right and negative to the left.

Since you have the equation for s(t) you can compute s(5) and answer part d immediately.

The derivative of s with respect to t is the velocity of s, call it v(t). If v is positive, s is going toward the right; if negative, toward the left; and if 0, then it just sits there not moving.

Since you have the equation for s, you can determine the equation for v(t).

The derivative of v with respect to t is the acceleration of s, call it a(t).

Since you've computed the equation of v(t), you can determine the equation for a(t)

With this, you can answer part b.

Now suppose s is going to change direction. Then at the point it changes, the velocity has to be 0. This isn't true the other way - the velocity could be 0 and s could continue in the same direction.

Still, the first step is to find all those values of t for which v(t) is 0. (There should be two)

For each T such that v(T) = 0, we need to compute a(t). If a(t) is positive, then s is going to move to the right from that point on. If a(t) is negative, then s is going to move to the left from that point.

This gives us enough to answer part a.

Given that, you can answer part c by adding all the individual pieces.