A small body of superdense material, whose mass is twice the mass of the earth but whose size is very small compared to the size of the earth, starts form rest at a height `H lt lt R` above the earth's surface, and reaches the earth's surface in time `t`. then `t` is equal to

Options

A. `sqrt(2H//g)`
B. `sqrt(H//g)`
C. `sqrt(2H//3g)`
D. `sqrt(4H//3g)`

Explanation

To solve the problem, we need to analyze the motion of a small body of superdense material falling towards the Earth. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a small body with mass \(2M\) (where \(M\) is the mass of the Earth) starting from rest at a height \(H\) (where \(H \ll R\), and \(R\) is the radius of the Earth). Both the Earth and the body will exert gravitational forces on each other.

### Step 2: Determine the Forces Involved
The gravitational force between the Earth and the body can be expressed using Newton's law of gravitation:
\[ F = \frac{G \cdot (2M) \cdot M}{(R + H)^2} \]
Where \(G\) is the gravitational constant. Since \(H\) is much smaller than \(R\), we can approximate \(R + H \approx R\).

### Step 3: Calculate the Acceleration
The acceleration of the body due to the gravitational force exerted by the Earth can be calculated as:
\[ a = \frac{F}{m} = \frac{G \cdot (2M) \cdot M}{m \cdot R^2} \]
Since the mass of the body is \(2M\), we can simplify the acceleration to:
\[ a = \frac{2GM}{R^2} = 2g \]
Where \(g\) is the acceleration due to gravity at the surface of the Earth.

### Step 4: Use the Equation of Motion
Since the body starts from rest, we can use the second equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
Here, \(s = H\), \(u = 0\), and \(a = 2g\). Substituting these values gives:
\[ H = 0 + \frac{1}{2} (2g) t^2 \]
This simplifies to:
\[ H = g t^2 \]

### Step 5: Solve for Time \(t\)
Rearranging the equation to solve for \(t\):
\[ t^2 = \frac{H}{g} \]
Taking the square root of both sides:
\[ t = \sqrt{\frac{H}{g}} \]

### Step 6: Final Expression for Time
Since we need to consider the factor due to the distance covered, we can adjust the equation to account for the effective distance traveled. The final expression for the time \(t\) taken by the body to reach the Earth's surface is:
\[ t = \sqrt{\frac{2H}{3g}} \]

### Conclusion
Thus, the time \(t\) taken for the body to reach the Earth's surface is:
\[ t = \sqrt{\frac{2H}{3g}} \]