(a)

The initial velocity is 

Let the position of a object at time is .

When an object is projected upwards, \ \

Acceleration is equal to acceleration of gravity .

Position at time is .

Then velocity of the object is the rate of change in distance .

Apply derivative on each side.

Graph

Velocity of the object at time is .

(b)

Position at time is .

Velocity of the object at time is .

To find the maximum height,

Substitute in the position function.

Maximum height is .

(c)

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

Integrate on both sides.

We know that when , then .

Therefore velocity function is .

(d)

velocity function is

Substitute in the velocity function.

Graph

At maximum height, Velocity is zero.

Therefore at , the height of the object is maximum.

For simplified, we make use of Simpson rule.

Let .

.

.

Therefore maximum height when resistance is considered is

(f)

We can observe that,

Maximum height when air resistance is not considered is .

Maximum height when air resistance is considered is .

Maximum height decreases when air resistance is considered.

(a) Velocity of the object at time is .

(b) Maximum height when air resistance is not considered .

(c) Velocity function is .

(d) The height of the object is maximum at .

(e) Maximum height when resistance is considered is

(f) Maximum height decreases when air resistance is considered.