Air offers a retardation of `6m//s^(2)` to a particle. If it is projected vertically upwards with a speed of `96m//s` from ground then the time of flight is : `(g = 10 m//s^(2))`

Options

A. `19.2s`
B. `12s`
C. `18s`
D. `24s`

Explanation

To solve the problem of a particle projected vertically upwards with an initial speed of 96 m/s and experiencing a retardation of 6 m/s² due to air, we can break the solution into two parts: the upward motion and the downward motion.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Initial velocity (u) = 96 m/s (upwards)
- Retardation due to air (a) = -6 m/s² (acting downwards)
- Acceleration due to gravity (g) = 10 m/s² (acting downwards)

2. **Calculate the Effective Acceleration During Upward Motion:**
- The total effective acceleration when the particle is moving upwards is the sum of gravitational acceleration and the retardation due to air:
\[ a_{\text{up}} = g + a = 10 + 6 = 16 \, \text{m/s}^2 \]

3. **Use the First Equation of Motion to Find Time to Reach Maximum Height:**
- At maximum height, the final velocity (v) will be 0 m/s. Using the equation:
\[ v = u + at \]
Substituting the known values:
\[ 0 = 96 - 16t_1 \implies 16t_1 = 96 \implies t_1 = \frac{96}{16} = 6 \, \text{s} \]

4. **Calculate the Maximum Height Reached (h):**
- Using the second equation of motion:
\[ h = ut + \frac{1}{2} a t^2 \]
Substituting the values:
\[ h = 96 \times 6 - \frac{1}{2} \times 16 \times (6^2) \]
\[ h = 576 - \frac{1}{2} \times 16 \times 36 = 576 - 288 = 288 \, \text{m} \]

5. **Calculate Time of Flight During Downward Motion:**
- When the particle comes down, the initial velocity (u) = 0 m/s, and it falls from a height of 288 m. The effective acceleration during the downward motion is:
\[ a_{\text{down}} = g - a = 10 - 6 = 4 \, \text{m/s}^2 \]
- Using the second equation of motion again:
\[ s = ut + \frac{1}{2} a t^2 \]
Substituting the values:
\[ 288 = 0 + \frac{1}{2} \times 4 \times t_2^2 \]
\[ 288 = 2t_2^2 \implies t_2^2 = \frac{288}{2} = 144 \implies t_2 = 12 \, \text{s} \]

6. **Calculate Total Time of Flight:**
- The total time of flight (T) is the sum of the time taken to go up (t1) and the time taken to come down (t2):
\[ T = t_1 + t_2 = 6 + 12 = 18 \, \text{s} \]

### Final Answer:
The total time of flight is **18 seconds**.