The acceleration due to gravity at a height 1 km above the earth is the same as at a depth d below the surface of earth

Options

A. d = 1 km
B. `d = (3)/(2) km`
C. d = 2 km
D. `d = (1)/(2)` km

Explanation

To solve the problem, we need to find the depth \( d \) below the Earth's surface where the acceleration due to gravity is the same as at a height of 1 km above the Earth's surface.

### Step-by-Step Solution:

1. **Understand the formulas for gravitational acceleration**:
- The acceleration due to gravity at a height \( h \) above the Earth's surface is given by:
\[ g_h = g \left(1 - \frac{2h}{R}\right) \]
- The acceleration due to gravity at a depth \( d \) below the Earth's surface is given by:
\[ g_d = g \left(1 - \frac{d}{R}\right) \]
where \( g \) is the acceleration due to gravity at the surface of the Earth, and \( R \) is the radius of the Earth.

2. **Set the two expressions equal**:
Since we want to find the depth \( d \) where the gravitational acceleration is the same at a height of 1 km, we set the two equations equal:
\[ g \left(1 - \frac{2h}{R}\right) = g \left(1 - \frac{d}{R}\right) \]

3. **Cancel \( g \) from both sides**:
Since \( g \) is a common factor, we can cancel it out:
\[ 1 - \frac{2h}{R} = 1 - \frac{d}{R} \]

4. **Substitute \( h = 1 \text{ km} = 1000 \text{ m} \)**:
Substitute \( h \) in the equation:
\[ 1 - \frac{2 \times 1000}{R} = 1 - \frac{d}{R} \]

5. **Simplify the equation**:
By simplifying, we can eliminate the 1s:
\[ -\frac{2 \times 1000}{R} = -\frac{d}{R} \]

6. **Multiply through by \( -R \)**:
This gives us:
\[ 2 \times 1000 = d \]

7. **Calculate \( d \)**:
Thus, we find:
\[ d = 2000 \text{ m} = 2 \text{ km} \]

### Conclusion:
The depth \( d \) below the Earth's surface where the acceleration due to gravity is the same as at a height of 1 km above the surface is \( 2 \text{ km} \).