At what height above the earth's surface, the value of g is same as that at a depth of 100 km ?

Explanation

To solve the problem of finding the height above the Earth's surface where the value of g is the same as that at a depth of 100 km, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Variation of g**:
- The acceleration due to gravity (g) varies with height (h) above the Earth's surface and depth (d) below the Earth's surface.
- The formula for g at height h above the surface is:
\[ g' = g_0 \left(1 - \frac{2h}{R}\right) \]
- The formula for g at depth d below the surface is:
\[ g' = g_0 \left(1 - \frac{d}{R}\right) \]
- Here, \(g_0\) is the acceleration due to gravity at the surface of the Earth, and \(R\) is the radius of the Earth.

2. **Setting Up the Equation**:
- Since we want to find the height where g is the same at both conditions, we can set the two equations equal to each other:
\[ g_0 \left(1 - \frac{2h}{R}\right) = g_0 \left(1 - \frac{d}{R}\right) \]

3. **Cancelling g_0**:
- We can cancel \(g_0\) from both sides since it is a common factor (assuming \(g_0 \neq 0\)):
\[ 1 - \frac{2h}{R} = 1 - \frac{d}{R} \]

4. **Simplifying the Equation**:
- By simplifying the equation, we get:
\[ -\frac{2h}{R} = -\frac{d}{R} \]
- This leads to:
\[ 2h = d \]

5. **Substituting the Given Depth**:
- We know from the problem that \(d = 100 \text{ km}\):
\[ 2h = 100 \text{ km} \]

6. **Solving for h**:
- Dividing both sides by 2 gives:
\[ h = \frac{100 \text{ km}}{2} = 50 \text{ km} \]

### Final Answer:
The height above the Earth's surface where the value of g is the same as that at a depth of 100 km is **50 km**.

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