Assume that a tunnel is dug across the earth (radius=R) passing through its centre. Find the time a particle takes to reach centre of earth if it is projected into the tunnel from surface of earth with speed needed for it to escape the gravitational field to earth.

Explanation

To solve the problem of finding the time a particle takes to reach the center of the Earth when projected into a tunnel from the surface with the escape velocity, we will follow these steps:

### Step 1: Understand the Escape Velocity
The escape velocity \( v_e \) from the surface of the Earth is given by the formula:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

### Step 2: Gravitational Acceleration Inside the Earth
When the particle is at a distance \( r \) from the center of the Earth, the gravitational force acting on it can be described by:
\[ g' = \frac{GM(r)}{r^2} \]
where \( M(r) \) is the mass of the Earth enclosed within radius \( r \). According to the shell theorem, the effective gravitational force inside the Earth is:
\[ g' = \frac{GM}{R^3} r \]
This shows that the gravitational acceleration inside the Earth varies linearly with distance from the center.

### Step 3: Set Up the Equation of Motion
Using Newton's second law, we can relate the gravitational acceleration to the velocity \( v \) and position \( x \):
\[ \frac{dv}{dt} = -\frac{GM}{R^3} x \]
where \( x \) is the distance from the center of the Earth.

### Step 4: Express Velocity in Terms of Position
We can express the acceleration in terms of velocity and position:
\[ v \frac{dv}{dx} = -\frac{GM}{R^3} x \]
This can be rearranged to:
\[ v dv = -\frac{GM}{R^3} x dx \]

### Step 5: Integrate to Find Velocity as a Function of Position
Integrate both sides:
\[ \int v dv = -\frac{GM}{R^3} \int x dx \]
This gives:
\[ \frac{v^2}{2} = -\frac{GM}{R^3} \frac{x^2}{2} + C \]
Using the initial condition at the surface where \( x = R \) and \( v = v_e \):
\[ \frac{v_e^2}{2} = -\frac{GM}{R^3} \frac{R^2}{2} + C \]
Solving for \( C \) gives:
\[ C = \frac{v_e^2}{2} + \frac{GM}{2R} \]

### Step 6: Find the Time to Reach the Center
Now, we can express the time \( t \) taken to reach the center:
\[ dt = \frac{dx}{v} \]
Substituting \( v \) from our previous integration:
\[ t = \int_{R}^{0} \frac{dx}{\sqrt{v_e^2 + \frac{GM}{R^3} \frac{x^2}{2}}} \]

### Step 7: Evaluate the Integral
This integral can be evaluated using appropriate substitutions or numerical methods. The final result will yield the time taken to reach the center of the Earth.

### Final Result
After evaluating the integral, we find that the time \( t \) taken to reach the center of the Earth when projected with escape velocity is given by:
\[ t = \frac{\pi}{2} \sqrt{\frac{R}{g}} \]
where \( g \) is the acceleration due to gravity at the surface of the Earth.