A small body of mass m falls to the earth from infintie distance away. What will be its velocity or reaching the earth? (Radius of the earth = R, acceleration due to gravity on the surface of the earth is g) :-

Options

A. `gR`
B. 2gR
C. `sqrt(gR)`
D. `sqrt(2gR)`

Answer

To find the velocity of a small body of mass \( m \) falling to the Earth from an infinite distance, we can use the principle of conservation of mechanical energy. Here’s a step-by-step solution:

### Step 1: Understand the Initial and Final States
- **Initial State**: The body is at an infinite distance from the Earth. At this point, both the potential energy and kinetic energy are zero.
\[ U_i = 0, \quad K_i = 0 \]

- **Final State**: The body reaches the surface of the Earth. At this point, it has potential energy and kinetic energy.
\[ U_f = -\frac{G M m}{R}, \quad K_f = \frac{1}{2} m v^2 \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

### Step 2: Apply Conservation of Mechanical Energy
According to the conservation of mechanical energy:
\[ U_i + K_i = U_f + K_f \]
Substituting the values:
\[ 0 + 0 = -\frac{G M m}{R} + \frac{1}{2} m v^2 \]

### Step 3: Rearranging the Equation
Rearranging the equation gives:
\[ \frac{1}{2} m v^2 = \frac{G M m}{R} \]

### Step 4: Cancel the Mass \( m \)
Since \( m \) is present on both sides of the equation, we can cancel it out (assuming \( m \neq 0 \)):
\[ \frac{1}{2} v^2 = \frac{G M}{R} \]

### Step 5: Solve for Velocity \( v \)
Multiplying both sides by 2:
\[ v^2 = \frac{2 G M}{R} \]
Taking the square root gives:
\[ v = \sqrt{\frac{2 G M}{R}} \]

### Step 6: Relate \( G \) to \( g \)
We know that the acceleration due to gravity \( g \) at the surface of the Earth is given by:
\[ g = \frac{G M}{R^2} \]
From this, we can express \( G M \) in terms of \( g \) and \( R \):
\[ G M = g R^2 \]

### Step 7: Substitute \( G M \) in the Velocity Equation
Substituting \( G M \) back into the velocity equation:
\[ v = \sqrt{\frac{2 g R^2}{R}} = \sqrt{2 g R} \]

### Final Answer
Thus, the velocity of the body when it reaches the surface of the Earth is:
\[ v = \sqrt{2 g R} \]