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 of finding the sum of the two times (t1 and t2) at which a projectile is at the same height, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Projectile Motion**:
A projectile follows a parabolic trajectory. It reaches the same height at two different times during its flight: once while ascending and once while descending.

2. **Time of Flight**:
The total time of flight (T) for a projectile launched vertically with an initial velocity \( u_y \) is given by the formula:
\[ T = \frac{2u_y}{g} \]
where \( g \) is the acceleration due to gravity.

3. **Identifying the Heights**:
Let’s denote the height at which the projectile is at the same height as \( h \). The projectile will reach this height at two different times, \( t_1 \) and \( t_2 \).

4. **Using the Equation of Motion**:
The vertical motion can be described by the equation:
\[ h = u_y t - \frac{1}{2} g t^2 \]
We can set this equation for both times \( t_1 \) and \( t_2 \):
\[ h = u_y t_1 - \frac{1}{2} g t_1^2 \]
\[ h = u_y t_2 - \frac{1}{2} g t_2^2 \]

5. **Setting Up the Equations**:
Rearranging both equations gives us:
\[ \frac{1}{2} g t_1^2 - u_y t_1 + h = 0 \]
\[ \frac{1}{2} g t_2^2 - u_y t_2 + h = 0 \]

6. **Using the Quadratic Formula**:
The above equations are quadratic in form. The sum of the roots of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( -\frac{b}{a} \). For our equations:
- \( a = \frac{1}{2} g \)
- \( b = -u_y \)

Thus, the sum of the times \( t_1 + t_2 \) is:
\[ t_1 + t_2 = -\frac{-u_y}{\frac{1}{2} g} = \frac{2u_y}{g} \]

7. **Relating to Time of Flight**:
From our earlier step, we know that:
\[ T = \frac{2u_y}{g} \]
Therefore, we can conclude that:
\[ t_1 + t_2 = T \]

### Final Answer:
The sum of the two times \( t_1 + t_2 \) for which the projectile is at the same height is equal to the total time of flight \( T \).

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