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

Options

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

Answer

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

### Step-by-step Solution:

1. **Understanding the Problem**:
We need to compare the acceleration due to gravity at a height \( h = 1 \text{ km} \) above the Earth's surface and at a depth \( d \) below the Earth's surface.

2. **Formula for Acceleration due to Gravity at Height**:
The acceleration due to gravity at a height \( h \) above the Earth's surface is given by:
\[ g_h = g \left(1 - \frac{2h}{R}\right) \]
where \( g \) is the acceleration due to gravity at the surface of the Earth and \( R \) is the radius of the Earth.

3. **Substituting the Height**:
For \( h = 1 \text{ km} = 1000 \text{ m} \), we can substitute this value into the formula:
\[ g_h = g \left(1 - \frac{2 \times 1000}{R}\right) \]

4. **Formula for Acceleration due to Gravity at Depth**:
The acceleration due to gravity at a depth \( d \) below the Earth's surface is given by:
\[ g_d = g \left(1 - \frac{d}{R}\right) \]

5. **Setting the Two Equations Equal**:
Since the problem states that these two accelerations are equal, we set the equations equal to each other:
\[ g \left(1 - \frac{2 \times 1000}{R}\right) = g \left(1 - \frac{d}{R}\right) \]

6. **Cancelling \( g \)**:
We can cancel \( g \) from both sides (assuming \( g \neq 0 \)):
\[ 1 - \frac{2 \times 1000}{R} = 1 - \frac{d}{R} \]

7. **Simplifying the Equation**:
Rearranging gives:
\[ -\frac{2 \times 1000}{R} = -\frac{d}{R} \]
Multiplying through by \( -R \) (assuming \( R > 0 \)):
\[ 2 \times 1000 = d \]

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

9. **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 Earth's surface is \( 2 \text{ km} \).

### Final Answer:
The depth \( d \) is \( 2 \text{ km} \).

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