A rocket is fired vertically up from the ground with a resultant vertical acceleration of `10 m//s^2.` The fuel is finished in 1 min and it continues to move up. (a) What is the maximum height reached? (b) Afte2r how much time from then will the maximum height be reached?(Take `g= 10 m//s^2`)

Explanation

To solve the problem step by step, we will break it down into two parts: (a) calculating the maximum height reached by the rocket and (b) determining the time taken to reach that maximum height after the fuel is exhausted.

### Part (a): Maximum Height Reached

1. **Identify Initial Conditions**:
- The rocket is fired with an initial vertical acceleration of \(10 \, \text{m/s}^2\).
- The fuel lasts for \(1 \, \text{minute} = 60 \, \text{seconds}\).

2. **Calculate the Height During Fuel Burn**:
- The initial velocity \(u = 0\) (since it starts from rest).
- Using the equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
- Substitute the values:
\[ h_1 = 0 \cdot 60 + \frac{1}{2} \cdot 10 \cdot (60)^2 \]
- Calculate:
\[ h_1 = \frac{1}{2} \cdot 10 \cdot 3600 = 18000 \, \text{meters} \]

3. **Calculate the Velocity at the End of Fuel Burn**:
- Using the equation \(v = u + at\):
\[ v = 0 + 10 \cdot 60 = 600 \, \text{m/s} \]

4. **Calculate the Additional Height After Fuel Exhaustion**:
- After the fuel is exhausted, the only force acting on the rocket is gravity, which decelerates it at \(g = 10 \, \text{m/s}^2\).
- The rocket will continue to rise until its velocity becomes \(0\).
- Using the equation \(v^2 = u^2 + 2as\):
\[ 0 = (600)^2 + 2 \cdot (-10) \cdot h_2 \]
- Rearranging gives:
\[ 0 = 360000 - 20h_2 \implies 20h_2 = 360000 \implies h_2 = 18000 \, \text{meters} \]

5. **Calculate the Total Maximum Height**:
- The total maximum height \(H_{max}\) is the sum of \(h_1\) and \(h_2\):
\[ H_{max} = h_1 + h_2 = 18000 + 18000 = 36000 \, \text{meters} = 36 \, \text{km} \]

### Part (b): Time to Reach Maximum Height After Fuel Exhaustion

1. **Calculate Time Taken to Reach Maximum Height After Fuel Exhaustion**:
- Use the equation \(v = u + at\):
\[ 0 = 600 - 10t \]
- Rearranging gives:
\[ 10t = 600 \implies t = 60 \, \text{seconds} \]

### Final Answers:
(a) The maximum height reached by the rocket is \(36 \, \text{km}\).
(b) The time taken to reach the maximum height after the fuel is exhausted is \(60 \, \text{seconds}\).