The relation between the time of flight of projectile `T_(f)` and the time to reach the maximum height `t_(m)` is

Options

A. `T_(f) = 2t_(m)`
B. `T_(f) = t_(m)`
C. `T_(f) = (t_(m))/(2)`
D. `T_(f) = sqrt(2)(t_(m))`

Explanation

To derive the relation between the time of flight of a projectile \( T_f \) and the time to reach the maximum height \( t_m \), we can follow these steps:

### Step 1: Understand the Motion of a Projectile
A projectile is launched at an angle with an initial velocity. It follows a parabolic trajectory, reaching a maximum height before descending back to the ground.

### Step 2: Define Time of Flight and Time to Maximum Height
- **Time of Flight \( T_f \)**: This is the total time the projectile is in the air, from launch until it returns to the ground.
- **Time to Reach Maximum Height \( t_m \)**: This is the time taken for the projectile to ascend from its launch point to the highest point of its trajectory.

### Step 3: Analyze the Motion
When the projectile is launched:
- It takes time \( t_m \) to reach the maximum height.
- After reaching the maximum height, it takes the same amount of time \( t_m \) to descend back to the ground.

### Step 4: Establish the Relationship
Since the time taken to ascend to the maximum height is equal to the time taken to descend back to the ground, we can express the total time of flight \( T_f \) as:
\[ T_f = t_m + t_m = 2t_m \]

### Step 5: Conclusion
Thus, the relationship between the time of flight \( T_f \) and the time to reach maximum height \( t_m \) is:
\[ T_f = 2t_m \]

### Final Answer
The relation between the time of flight of a projectile \( T_f \) and the time to reach the maximum height \( t_m \) is:
\[ T_f = 2t_m \]

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