The function is .

(a)

Velocity at time  :

The motion of the particle is , where  is in seconds and  is measured in feet.

Velocity function is derivative of the position function and is defined as .

Velocity at time  is .

(b)

Find the velocity after  sec:

Velocity function is .

Substitute  in above expression.

Velocity after  seconds is  ft/sec.

(c)

Find the time when particle is at rest.

Velocity function is .

When the particle is at rest, the initial velocity is zero.

 and .

The particle is at rest when  sec and  sec.

(d)

Find the time when the particle move in positive direction.

Velocity function is .

When the particle is at rest, the initial velocity should be greater than zero.

The above inequality is true, when both factors are positive or both factors are negative.

If both factors are positive, then  and .

It is concluded that .

The particle moves in positive direction when .

(e)

Find the distance traveled by particle in  sec.

The time intervals are  and  from (d).

Find the distance traveled by the particle in the interval .

The position function is .

.

.

Find the distance traveled by the particle in the interval .

.

Total distance traveled in  sec is  ft.

(f)

The schematic diagram of motion of the particle.

(g)

Find the acceleration at time  and after  sec.

Acceleration is derivative of the velocity function.

Velocity function is .

Acceleration after  sec:

Acceleration after  sec is  .

(h)

Graph :

Graph the position, velocity and acceleration functions for .

.

(i)

The particle speeds up when the velocity is positive and increasing.

Thus, from the graph it happens in the interval  .

The particle also speeds up when the velocity is negative and decreasing.

Thus, from the graph it happens in the interval .

The particle slows down when the velocity and acceleration have opposite signs.

Thus, from the graph it happens in the interval  . 

(a) Velocity at time  is  .

(b) Velocity after 3 seconds is given by  ft/sec.

(c) The particle is at rest at  sec and  sec .

(d) The particle moves in positive direction when .

(e) Total distance traveled in 8 sec is  ft.

(g) Acceleration after 3 sec is  .

(h)

(i) 

When the particle speeds up in the interval  or .

When the particle slows down in the interval .