A rocket, initially at rest, is fired vertically with an upward acceleration of `10.0m//s^(2)` . At an altitude of 0.500 km, the engine of the rocket cuts off. What is the maximum altitude it achieves ?

Answer

To solve the problem step by step, we will follow the outlined process:

### Step 1: Understand the problem
The rocket is initially at rest and is fired vertically with an upward acceleration of \(10.0 \, \text{m/s}^2\). It reaches an altitude of \(0.500 \, \text{km}\) (which is \(500 \, \text{m}\)) before the engine cuts off. We need to find the maximum altitude it achieves after the engine cuts off.

### Step 2: Convert the altitude
Convert the altitude from kilometers to meters:
\[ 0.500 \, \text{km} = 0.500 \times 1000 = 500 \, \text{m} \]

### Step 3: Calculate the velocity at the cutoff point
Use the kinematic equation to find the velocity of the rocket at the altitude of \(500 \, \text{m}\):
\[ v^2 = u^2 + 2as \]
Where:
- \(u = 0 \, \text{m/s}\) (initial velocity, since it starts from rest)
- \(a = 10 \, \text{m/s}^2\) (upward acceleration)
- \(s = 500 \, \text{m}\) (distance traveled)

Plugging in the values:
\[ v^2 = 0 + 2 \times 10 \times 500 \]
\[ v^2 = 10000 \]
\[ v = \sqrt{10000} = 100 \, \text{m/s} \]

### Step 4: Analyze the motion after the engine cuts off
Once the engine cuts off, the rocket will move upward until it reaches its maximum height, at which point its velocity will be \(0 \, \text{m/s}\). The only force acting on it is gravity, which decelerates it at \(10 \, \text{m/s}^2\).

### Step 5: Calculate the additional height gained after cutoff
Use the kinematic equation for the upward motion after the engine cuts off:
\[ v^2 = u^2 + 2as \]
Where:
- \(v = 0 \, \text{m/s}\) (final velocity at maximum height)
- \(u = 100 \, \text{m/s}\) (initial velocity at cutoff)
- \(a = -10 \, \text{m/s}^2\) (acceleration due to gravity, negative because it is downward)

Plugging in the values:
\[ 0 = (100)^2 + 2 \times (-10) \times s \]
\[ 0 = 10000 - 20s \]
\[ 20s = 10000 \]
\[ s = \frac{10000}{20} = 500 \, \text{m} \]

### Step 6: Calculate the maximum altitude
The maximum altitude is the sum of the altitude at cutoff and the additional height gained:
\[ \text{Maximum altitude} = 500 \, \text{m} + 500 \, \text{m} = 1000 \, \text{m} \]

### Final Answer
The maximum altitude achieved by the rocket is:
\[ \text{Maximum altitude} = 1000 \, \text{m} = 1 \, \text{km} \]

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