Analysis of the problem:

The problem states that a body is thrown vertically upwards from the top of a tower. Let's denote the height of the tower as 'h'. It is given that when the body is thrown upwards, it reaches the ground in 't1' seconds. Similarly, when the body is thrown downwards from the same height with the same speed, it reaches the ground in 't2' seconds. We need to find the time it takes for the body to reach the ground when it falls freely from the top of the tower.

Solution:

1. Time taken to reach the ground when thrown upwards:

When the body is thrown upwards, it follows a projectile motion. The initial velocity of the body is positive, and the acceleration due to gravity acts in the opposite direction. At its highest point, the body momentarily comes to rest and then starts falling downwards. The total time taken by the body to reach the ground can be divided into two parts:

- The time taken to reach the highest point (going upwards)
- The time taken to reach the ground from the highest point (coming downwards)

Since the body is thrown upwards with the same speed as when it is thrown downwards, the time taken to reach the highest point (going upwards) is equal to the time taken to reach the ground from the highest point (coming downwards). Therefore, the total time taken to reach the ground when thrown upwards is 't1'.

2. Time taken to reach the ground when thrown downwards:

When the body is thrown downwards, it also follows a projectile motion. The initial velocity of the body is negative, and the acceleration due to gravity acts in the same direction. The time taken to reach the ground when thrown downwards is given as 't2'.

3. Time taken to reach the ground when falling freely:

When the body is allowed to fall freely from the top of the tower, it is not subjected to any initial velocity. It only experiences the acceleration due to gravity acting in the downward direction. In this case, the time taken to reach the ground can be found using the equation of motion:

h = (1/2) * g * t^2

where 'h' is the height of the tower and 'g' is the acceleration due to gravity. We need to solve this equation for 't' to find the time taken to reach the ground.

4. Conclusion:

To summarize, when the body is thrown upwards or downwards, it takes 't1' and 't2' seconds, respectively, to reach the ground. However, when it falls freely from the top of the tower, the time taken to reach the ground can be found using the equation of motion. By solving the equation, we can determine the time it takes for the body to reach the ground when falling freely.

T=t1*t2