A rocket is fired vertically from the ground. It moves upwards with a constant acceleration `10m//s^(2)` after 30 sec the fuel is finished. After what time from the instant of firing the rocket will attain the maximum height? `g=10m//s^(2)`:-

Options

A. 30s
B. 45s
C. 60s
D. 75s

Explanation

To solve the problem of determining the time at which the rocket will attain its maximum height after being fired vertically from the ground, we can break it down into two main phases: the powered ascent and the free-fall ascent.

### Step-by-Step Solution:

1. **Determine the velocity at the end of the powered ascent:**
- Given:
- Initial velocity, \( u = 0 \, \text{m/s} \) (since the rocket starts from rest)
- Constant acceleration, \( a = 10 \, \text{m/s}^2 \)
- Time of powered ascent, \( t_1 = 30 \, \text{seconds} \)
- Using the kinematic equation:
\[ v = u + at \]
Substituting the given values:
\[ v = 0 + (10 \, \text{m/s}^2 \times 30 \, \text{seconds}) = 300 \, \text{m/s} \]
- So, the velocity at the end of the powered ascent is \( 300 \, \text{m/s} \).

2. **Determine the time taken to reach the maximum height after the fuel is finished:**
- After the fuel is finished, the rocket will continue to move upwards under the influence of gravity alone.
- Given:
- Initial velocity for this phase, \( u = 300 \, \text{m/s} \)
- Final velocity at maximum height, \( v = 0 \, \text{m/s} \)
- Acceleration due to gravity, \( g = -10 \, \text{m/s}^2 \) (negative because it acts downward)
- Using the kinematic equation:
\[ v = u + at \]
Substituting the values:
\[ 0 = 300 \, \text{m/s} + (-10 \, \text{m/s}^2) \times t_2 \]
Solving for \( t_2 \):
\[ 0 = 300 - 10t_2 \]
\[ 10t_2 = 300 \]
\[ t_2 = \frac{300}{10} = 30 \, \text{seconds} \]

3. **Calculate the total time to reach the maximum height:**
- The total time \( T \) from the instant of firing to the maximum height is the sum of the time during the powered ascent and the time during the free-fall ascent:
\[ T = t_1 + t_2 = 30 \, \text{seconds} + 30 \, \text{seconds} = 60 \, \text{seconds} \]

### Final Answer:
The rocket will attain its maximum height 60 seconds after being fired.