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 of finding the speed of a rocket fired from the Earth's surface at interstellar space, we will use the principle of conservation of mechanical energy. Here are the steps to solve the problem:

### Step-by-Step Solution:

1. **Understanding Initial Conditions**:
- The rocket is fired 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. **Total Mechanical Energy at the Earth's Surface**:
- The total mechanical energy \( E_i \) at the Earth's surface is the sum of kinetic energy (KE) and gravitational potential energy (PE).
- Kinetic Energy (KE) at the surface:
\[ KE_i = \frac{1}{2} m u^2 = \frac{1}{2} m (3\sqrt{gR})^2 = \frac{1}{2} m \cdot 9gR = \frac{9}{2} mgR \]
- Gravitational Potential Energy (PE) at the surface:
\[ PE_i = -\frac{GMm}{R} = -mgR \quad (\text{where } G \text{ is the gravitational constant and } M \text{ is the mass of the Earth}) \]
- Therefore, the total energy at the Earth's surface is:
\[ E_i = KE_i + PE_i = \frac{9}{2} mgR - mgR = \frac{9}{2} mgR - \frac{2}{2} mgR = \frac{7}{2} mgR \]

3. **Total Mechanical Energy in Interstellar Space**:
- In interstellar space, the gravitational potential energy is zero (PE = 0) because gravity is negligible.
- Let \( v \) be the speed of the rocket in interstellar space. The total mechanical energy \( E_f \) in interstellar space is:
\[ E_f = KE_f + PE_f = \frac{1}{2} mv^2 + 0 = \frac{1}{2} mv^2 \]

4. **Applying Conservation of Energy**:
- According to the conservation of mechanical energy, the total energy at the Earth's surface must equal the total energy in interstellar space:
\[ E_i = E_f \]
\[ \frac{7}{2} mgR = \frac{1}{2} mv^2 \]

5. **Solving for \( v \)**:
- Cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{7}{2} gR = \frac{1}{2} v^2 \]
- Multiply both sides by 2:
\[ 7gR = v^2 \]
- Taking the square root:
\[ v = \sqrt{7gR} \]

### Final Answer:
The speed of the rocket at interstellar space is:
\[ v = \sqrt{7gR} \]