A body at rest starts from a point at a distance r (gtR) from the centre of the Earth. If M and R stand for the speed of the body when it reaches the Earth surface is

Options

A. `sqrt((2GM)/(R ))`
B. `sqrt((2GM)/(R-r))`
C. `sqrt((2GM(r-R))/(Rr))`
D. Zero

Answer

To solve the problem of finding the speed of a body when it reaches the Earth's surface, we can use the principle of conservation of energy. Here's a step-by-step solution:

### Step 1: Understand the problem
A body starts from rest at a distance \( r \) (where \( r > R \), and \( R \) is the radius of the Earth) from the center of the Earth. We need to find the speed of the body when it reaches the surface of the Earth.

### Step 2: Write the conservation of energy equation
The total mechanical energy at the starting point (initial energy) must equal the total mechanical energy when the body reaches the surface of the Earth (final energy).

**Initial Energy:**
- Kinetic Energy (KE_initial) = 0 (since the body starts from rest)
- Gravitational Potential Energy (PE_initial) = \(-\frac{GMm}{r}\)

**Final Energy:**
- Kinetic Energy (KE_final) = \(\frac{1}{2}mv^2\) (where \( v \) is the speed at the Earth's surface)
- Gravitational Potential Energy (PE_final) = \(-\frac{GMm}{R}\)

### Step 3: Set up the equation
Using the conservation of energy:
\[ KE_{\text{initial}} + PE_{\text{initial}} = KE_{\text{final}} + PE_{\text{final}} \]
Substituting the energies:
\[ 0 - \frac{GMm}{r} = \frac{1}{2}mv^2 - \frac{GMm}{R} \]

### Step 4: Simplify the equation
Rearranging gives:
\[ -\frac{GMm}{r} + \frac{GMm}{R} = \frac{1}{2}mv^2 \]
Factoring out \( GMm \):
\[ GMm \left(\frac{1}{R} - \frac{1}{r}\right) = \frac{1}{2}mv^2 \]
Dividing both sides by \( m \) (assuming \( m \neq 0 \)):
\[ GM \left(\frac{1}{R} - \frac{1}{r}\right) = \frac{1}{2}v^2 \]

### Step 5: Solve for \( v^2 \)
Multiplying both sides by 2:
\[ 2GM \left(\frac{1}{R} - \frac{1}{r}\right) = v^2 \]

### Step 6: Solve for \( v \)
Taking the square root gives:
\[ v = \sqrt{2GM \left(\frac{1}{R} - \frac{1}{r}\right)} \]

### Final Result
The speed of the body when it reaches the Earth's surface is:
\[ v = \sqrt{2GM \left(\frac{1}{R} - \frac{1}{r}\right)} \]