**Example 1**

A ball of mass is thrown straight up from the ground with an initial velocity of . Ignoring air resistance, find the maximum height and the time the ball takes to attain the maximum height.

**Solution**

Let the height of the ball from the ground at any time be .

Since the force acting on the ball is , Newton's second law of motion, , gives the relation

Thus

Integrating both sides,

Velocity at , so yields

, this

Now, we have

Integrating ,

Since

The equation of position at any time is

To find time () at the maximum height, we set the first derivative of (eq3) equals to (set velocity = ) to have

, .

Check that the second derivative at this point is negative, i.e., .

Maximum height possible the ball can attain is

.

**Example 2**

A ball weighing is thrown vertically upwards from a point above the surface of the earth with an initial velocity of . As it rises it is acted upon by air resistance that is numerically equal to (in pounds), where is the velocity in feet per second. How high will the ball rise?

**Solution**

Let the height of the ball at at any time from the ground be , and assume that the upward direction is positive. From Newton's second law of motion , where is the mass of an object and is acceleration due to gravity,

,

Separating variables,

Integrating,

Applying ,

From

, where

At the maximum height, ,

So , where is the time the ball takes to reach maximum height.

Hence

To be continued.