A body of mass `m` at inifnity is approaching the centre of a hypothetical hollow planet of mass `M` and radius `R`. The speed of the body when it passes the centre of the planet through a diametrical tunnel is

Options

A. `sqrt((GM)/(R ))`
B. `sqrt((2GM)/(R ))`
C. Zero
D. none of these

Answer

To find the speed of a body of mass `m` when it passes through the center of a hypothetical hollow planet of mass `M` and radius `R`, we can use the principle of conservation of mechanical energy.

### Step-by-Step Solution:

1. **Understanding the System**:
- The body starts from rest at infinity, meaning its initial kinetic energy is zero.
- As it approaches the hollow planet, it will be influenced by gravitational potential energy.

2. **Initial Energy at Infinity**:
- At infinity, the gravitational potential energy (U) is defined as zero.
- The initial kinetic energy (K) is also zero since the body starts from rest.
- Therefore, the total initial mechanical energy (E_initial) is:
\[ E_{\text{initial}} = K + U = 0 + 0 = 0 \]

3. **Energy at the Center of the Planet**:
- When the body reaches the center of the hollow planet, it will have some kinetic energy (K) and gravitational potential energy (U).
- The kinetic energy at the center is given by:
\[ K = \frac{1}{2} mv^2 \]
- The gravitational potential energy at the center of a hollow planet is given by:
\[ U = -\frac{GMm}{R} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

4. **Conservation of Mechanical Energy**:
- According to the conservation of energy:
\[ E_{\text{initial}} = E_{\text{final}} \]
- Thus, we have:
\[ 0 = \frac{1}{2} mv^2 - \frac{GMm}{R} \]

5. **Solving for Velocity**:
- Rearranging the equation gives:
\[ \frac{1}{2} mv^2 = \frac{GMm}{R} \]
- Dividing both sides by \( m \) (assuming \( m \neq 0 \)):
\[ \frac{1}{2} v^2 = \frac{GM}{R} \]
- Multiplying both sides by 2:
\[ v^2 = \frac{2GM}{R} \]
- Taking the square root gives:
\[ v = \sqrt{\frac{2GM}{R}} \]

### Final Answer:
The speed of the body when it passes through the center of the planet is:
\[ v = \sqrt{\frac{2GM}{R}} \]