There are two values of time for which a projectile is at the same height. The sum of these two times is equal to (T = time of flight of the projectile)

Options

A. `(3T)/(2)`
B. `(4T)/(2)`
C. `(3T)/(4)`
D. T

Answer

To solve the problem, we need to analyze the motion of a projectile and understand the relationship between the time it takes to reach a certain height during its ascent and descent.

### Step-by-Step Solution:

1. **Understanding Projectile Motion**:
- A projectile follows a parabolic trajectory when it is launched. The height of the projectile at any time can be described by the equations of motion.

2. **Identifying the Times**:
- Let \( t_1 \) be the time taken to reach a certain height \( H \) during the upward motion.
- Let \( t_2 \) be the time taken to reach the same height \( H \) during the downward motion.

3. **Time of Flight**:
- The total time of flight \( T \) for a projectile is the time taken to go up and come back down to the same level from which it was launched.
- The projectile will take the same amount of time to ascend to height \( H \) as it will to descend back to that height.

4. **Relationship Between Times**:
- According to the properties of projectile motion, the time taken to reach a certain height during ascent and the time taken to return to that height during descent are related.
- Therefore, we can express this relationship as:
\[ t_1 + t_2 = T \]
- This means that the sum of the times \( t_1 \) and \( t_2 \) is equal to the total time of flight \( T \).

5. **Conclusion**:
- Thus, we conclude that for a projectile, the sum of the two times at which it reaches the same height is equal to the total time of flight.

### Final Answer:
The sum of the two times for which the projectile is at the same height is equal to the time of flight \( T \).

---