Let us analyze the equation "v = u + at" for the vertical motion under gravity with the help of an example. We consider a ball thrown upwards from ground with an initial speed of 30 m/s. In the frame of reference with upward direction as positive,

Vertical motion under gravity |
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Putting this value in the equation, we have :

v = 30 – 10 t

The important aspect of this equation is that velocity evaluates to both positive and negative values; positive for upward motion and negative for downward motion. The final velocity (v) is positive for t < 3 seconds, zero for t = 3 seconds and negative for t > 3 seconds. The total time taken for the complete up and down journey is 3 (for upward motion) + 3 (for downward motion) = 6 seconds.

The velocities of the ball at successive seconds are :

```
----------------------------------
Time (t) Final velocity (v)
in seconds in m/s
----------------------------------
0.0 30
1.0 20
2.0 10
3.0 0
4.0 -10
5.0 -20
6.0 -30
----------------------------------
```

The corresponding velocity – time plot looks like as shown in the figure.

Velocity – time plot |
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We notice following important characteristics of the motion :

1: The velocity at the maximum height is zero (v=0).

2: The time taken by the ball to reach maximum height is obtained as :

3: The ball completely regains its speed when it returns to ground, but the motion is directed in the opposite direction i.e.

4: The time taken for the complete round trip is :

The time taken for the complete journey is twice the time taken to reach the maximum height. It means that the ball takes equal time in upward and downward journey. Thus, the total motion can be considered to be divided in two parts of equal duration.

5: The velocity of the ball is positive in the first half of motion; Zero at the maximum height; negative in the second of the motion.

6: The velocity is decreasing all through the motion from a positive value to less positive value in the first half and from a less negative value to more negative value in the second half of the motion. This renders acceleration to be always negative (directed in -y direction), which is actually the case.

7: The velocity (positive) and acceleration (negative) in the first part are opposite in direction and the resulting speed is decreasing. On the other hand, the velocity (negative) and acceleration (negative) in the second part are in the same direction and the resulting speed is increasing.