At time t, the position of a particle moving along the s-axis is s=t^3-t^2-8t+5 on the on the interval [0,5].

Question

Find

a) When is the particle moving to the left? moving to the right?

b) The displacement during the first 3 seconds

c) The average velocity during the first 4 seconds

d) the instantaneous velocity when t=2 sec

e) The acceleration of the particle when t=5 sec

f) total distance traveled by the particle in 5 sec

g) When does the particle change direction

h) When is the particle at rest?

f) When is the particle speeding up? slowing down?

Answer 1

Graph the equation s(t)

There is a minimum at (2, -7)

The slope of the displacement curve is the velocity. When is the object moving left, when the velocity is negative (when the slope is negative) so on the interval [0, 2) then its moving right on the interval (2, 5]

The displacement after 3 sec is s(3) - s(0)

The average velocity for the first 4 sec = Δx/Δt = [s(4) - s(0)]/4

The instantaneous velocity requires you to take the derivative of the displacement function the to find the instantaneous velocity at t = 2 sec means find s'(2)

To find the acceleration, you'll need the second derivative

Question f is tricky. I think they mean what is the total displacement. If so, s(5) - s(0).

Particle is going left but slowing down between 0 and 2. At 2 it stops. Then it changes direction and goes right and goes faster and faster as time goes on