A rocket is fired with a speed `u=3sqrt(gR)` from the earth surface . What will be its speed at interstellar space ?

Options

A. zero
B. `sqrt(2gR)`
C. `sqrt(7gR)`
D. `sqrt(3gR)`

Answer

To solve the problem, we can use the principle of conservation of mechanical energy. The total mechanical energy (kinetic energy + potential energy) at the surface of the Earth will be equal to the total mechanical energy in interstellar space.

### Step-by-Step Solution:

1. **Identify the initial conditions**:
The rocket is fired from the Earth's surface with an initial speed \( u = 3\sqrt{gR} \), where \( g \) is the acceleration due to gravity at the Earth's surface, and \( R \) is the radius of the Earth.

2. **Write the expression for initial kinetic energy (KE) and gravitational potential energy (PE)**:
- Initial kinetic energy at the Earth's surface:
\[ KE_i = \frac{1}{2} m u^2 = \frac{1}{2} m (3\sqrt{gR})^2 = \frac{1}{2} m \cdot 9gR = \frac{9}{2} mgR \]
- Initial gravitational potential energy at the Earth's surface:
\[ PE_i = -\frac{GMm}{R} = -mgR \]
(where \( G \) is the universal gravitational constant and \( M \) is the mass of the Earth).

3. **Calculate the total mechanical energy at the Earth's surface**:
\[ E_i = KE_i + PE_i = \frac{9}{2} mgR - mgR = \frac{9}{2} mgR - \frac{2}{2} mgR = \frac{7}{2} mgR \]

4. **Consider the conditions in interstellar space**:
In interstellar space, the gravitational potential energy is considered to be zero (as we are far away from any massive body). Thus, the total mechanical energy in interstellar space is purely kinetic:
\[ E_f = KE_f + PE_f = \frac{1}{2} m v'^2 + 0 = \frac{1}{2} m v'^2 \]

5. **Set the total mechanical energy at the Earth's surface equal to that in interstellar space**:
\[ \frac{7}{2} mgR = \frac{1}{2} m v'^2 \]

6. **Cancel \( m \) from both sides** (assuming \( m \neq 0 \)):
\[ \frac{7}{2} gR = \frac{1}{2} v'^2 \]

7. **Multiply both sides by 2**:
\[ 7gR = v'^2 \]

8. **Take the square root to find \( v' \)**:
\[ v' = \sqrt{7gR} \]

### Final Answer:
The speed of the rocket in interstellar space will be \( v' = \sqrt{7gR} \).