A rocket is launched vertical from the surface of the earth of radius R with an initial speed `v`. If atmospheric resistance is neglected, then maximum height attained by the rocket is

Options

A. `h=(R)/(((2gR)/(v^(2))-1))`
B. `h=(R)/(((2gR)/(v^(2))+1))`
C. `h=(R^(2))/(((2gR)/(v^(2))-1))`
D. `h=(R^(2))/(((2gR)/(v^(2))+1))`

Explanation

To find the maximum height attained by the rocket launched vertically from the surface of the Earth, we can use the principle of conservation of energy.

### Step-by-Step Solution:

1. **Identify the Initial Conditions**:
- The rocket is launched from the surface of the Earth with an initial speed \( v \).
- The radius of the Earth is \( R \).
- We neglect atmospheric resistance.

2. **Write the Energy Conservation Equation**:
The total mechanical energy at the launch point (kinetic energy) will be equal to the total mechanical energy at the maximum height (potential energy).
\[ \text{Initial Kinetic Energy} = \text{Potential Energy at maximum height} \]
\[ \frac{1}{2} mv^2 = mg(h + R) \]
Here, \( h \) is the maximum height attained by the rocket above the Earth's surface.

3. **Rearranging the Equation**:
We can simplify the equation by canceling \( m \) (mass of the rocket) from both sides:
\[ \frac{1}{2} v^2 = g(h + R) \]
Now, we can isolate \( h \):
\[ h + R = \frac{v^2}{2g} \]
\[ h = \frac{v^2}{2g} - R \]

4. **Expressing \( h \) in Terms of \( R \)**:
To express \( h \) in a more useful form, we can factor out \( R \):
\[ h = \frac{v^2}{2g} - R \]

5. **Final Expression for Maximum Height**:
To express the height in a more standard form, we can rewrite it as:
\[ h = R\left(\frac{v^2}{2gR} - 1\right) \]

6. **Final Result**:
The maximum height attained by the rocket is:
\[ h = \frac{R}{2g} \left( \frac{v^2}{R} - 1 \right) \]