(a)

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

Find the velocity at time  :

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.

Position function is .

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

.

The particle is at rest when 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 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 .

.

.

Find the distance traveled by the particle in the interval .

.

Total distance is ft.

(f)

The schematic diagram of the motion of the particle:

(g)

Find the acceleration at time and after sec.

Acceleration is derivative of the velocity function.

Velocity function is .

The acceleration at time is .

Acceleration after sec :

Substitute in the acceleration.

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 slows down when the velocity and acceleration have opposite signs.

Thus, from the graph it happens in the interval and  .

(a) Velocity at time is .

(b) Velocity after seconds is ft/sec.

(c) The particle is at rest when sec.

(d) The particle moves in positive direction when .

(e) Total distance is ft.

(f) The schematic diagram of the motion of the particle is

(g) Acceleration is and after sec is  .

(h)

(i)

When the particle speeds up in the interval .

When the particle slows down in the interval and  .