(a)

An object is dropped from the height of .

Let the velocity of a object at time be .

Velocity at time is .

Initial velocity is zero.

Acceleration is equal to acceleration of gravity .

Graph :

Graph the initial velocity :

Velocity of the object at time is .

(b)

Velocity of the object at time is .

Velocity of the object is the rate of change in distance .

Integrate on both sides. 

At , height then .

Therefore the position function is .

(c)

If air resistance is considered then rate of change in velocity is

Integrate on both sides.

At , Velocity then .

Velocity function is .

(d)

Find .

Substitute in the function.

Therefore .

(e)

Velocity function is .

Integrate the velocity function.

At , height then .

.

Substitute in the velocity function.

.

Graph:

Graph the function and .

Time required for the object to reach the ground.

Time required when air resistance was neglected is .

Therefore additional time taken by air resistance is

Additional time required to reach ground when air resistance is considered is .

(f)

If  value is increased then the time taken by the object to reach ground decreases.

Consider .

Time taken by the object to reach ground when air resistance is not considered is .

Time taken by the object to reach ground when air resistance is considered is .

The time taken by the object to reach ground decreases when air resistance is considered.

(a) Velocity of the object at time is .

(b) The position function is .

(c) Velocity function when air resistance is considered is .

(d) .

(e) Additional time taken to reach ground .

(f) The time taken by the object to reach ground decreases when is increased.