A particle falls towards the earth from inifinity. The velocity with which it reaches the earth is surface is

Options

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

Answer

To find the velocity with which a particle reaches the surface of the Earth when falling from infinity, we can use the principle of conservation of mechanical energy. Here’s a step-by-step solution:

### Step 1: Understand the Initial Conditions
At infinity, the particle is at rest, so its initial kinetic energy (KE_initial) is zero, and the gravitational potential energy (PE_initial) is also considered to be zero.

- **Initial Kinetic Energy (KE_initial)** = 0
- **Initial Potential Energy (PE_initial)** = 0

### Step 2: Define the Final Conditions
When the particle reaches the surface of the Earth, it will have some kinetic energy and gravitational potential energy.

- **Final Kinetic Energy (KE_final)** = \( \frac{1}{2} m v^2 \)
- **Final Potential Energy (PE_final)** = \( -\frac{GMm}{R} \) (where \( G \) is the universal gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the particle, and \( R \) is the radius of the Earth)

### Step 3: Apply Conservation of Mechanical Energy
According to the conservation of mechanical energy:
\[ KE_{initial} + PE_{initial} = KE_{final} + PE_{final} \]

Substituting the values we have:
\[ 0 + 0 = \frac{1}{2} m v^2 - \frac{GMm}{R} \]

### Step 4: Rearranging the Equation
Rearranging the equation gives:
\[ \frac{1}{2} m v^2 = \frac{GMm}{R} \]

### Step 5: Cancel the Mass of the Particle
Since \( m \) appears on both sides, we can cancel it:
\[ \frac{1}{2} v^2 = \frac{GM}{R} \]

### Step 6: Solve for Velocity
Multiplying both sides by 2:
\[ v^2 = \frac{2GM}{R} \]

Taking the square root of both sides gives:
\[ v = \sqrt{\frac{2GM}{R}} \]

### Step 7: Relate \( g \) to \( G \)
We know that the acceleration due to gravity \( g \) at the surface of the Earth is given by:
\[ g = \frac{GM}{R^2} \]

Thus, we can express \( GM \) as:
\[ GM = gR^2 \]

### Step 8: Substitute Back into the Velocity Equation
Substituting \( GM \) back into the velocity equation:
\[ v = \sqrt{\frac{2gR^2}{R}} \]
\[ v = \sqrt{2gR} \]

### Final Answer
The velocity with which the particle reaches the surface of the Earth is:
\[ v = \sqrt{2gR} \]