The position function is , where is in seconds and is measured in feet.

(a)

Find 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.

Velocity function is .

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

If , then for all .

Therefore .

If , then .

If , then .

If , then .

The particle is at rest when 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.

is negative in III and  IV quadrants.

Therefore,

.

.

.

The particle moves in positive direction when .

(e)

Find the distance traveled by particle in sec.

The time intervals are and from (d).

The position function is .

Find the distance traveled by the particle in the interval .

.

.

Find the distance traveled by the particle in the interval .

.

.

Total distance traveled by the particle 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 is  and 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  and .

(a) Velocity at time is .

(b) Velocity after seconds is ft/sec.

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

(d) The particle moves in positive direction when .

(e) Total distance traveled  by the particle in sec is ft.

(f)

The schematic diagram of motion of the particle is

(g) Acceleration is  and after sec is .

(h)

(i)

When the particle speeds up in the interval or .

When the particle slows down in the interval or .