A particle is thrown vertically upwards from the surface of the earth. Let `T_(P)` be the time taken by the particle to travel from a point P above the earth to its highest point and back to the point P. Similarly, let `T_(Q)` be the time taken by the particle to travel from another point Q above the earth to its highest point and back to the same in terms of `T_(P), T_(Q)` and H, is :-

Explanation

To solve the problem, we need to analyze the motion of a particle thrown vertically upwards from two different points, P and Q, above the Earth's surface. We will derive the relationship between the times taken to reach the highest point and return to the starting points.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- When a particle is thrown vertically upwards, it travels to a maximum height and then returns back to its starting point. The time taken for this journey from point P to the highest point and back to P is denoted as \( T_P \).
- Similarly, for point Q, the time taken is denoted as \( T_Q \).

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