At what height from the surface of earth the gravitation potential and the value of g are `- 5 . 4 xx 10^(7) J kg^(-2) and 6 . 0 ms^(-2)` respectively. Take the radius of earth as 6400 km

Options

A. 2600 km
B. 1600 km
C. 1400 km
D. 2000 km

Explanation

To solve the problem, we need to determine the height \( h \) from the surface of the Earth where the gravitational potential \( V \) is \( -5.4 \times 10^7 \, \text{J/kg} \) and the value of \( g \) (acceleration due to gravity) is \( 6.0 \, \text{m/s}^2 \). The radius of the Earth \( R \) is given as \( 6400 \, \text{km} \).

### Step-by-Step Solution:

1. **Understanding the equations for gravitational potential and acceleration due to gravity**:
- The gravitational potential \( V \) at a distance \( r \) from the center of the Earth is given by:
\[ V = -\frac{GM}{r} \]
- The acceleration due to gravity \( g \) at a distance \( r \) from the center of the Earth is given by:
\[ g = \frac{GM}{r^2} \]
Here, \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

2. **Setting the distances**:
- Let \( h \) be the height above the Earth's surface. The distance from the center of the Earth to the point at height \( h \) is:
\[ r = R + h \]
where \( R = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} \).

3. **Substituting the values into the equations**:
- From the gravitational potential equation:
\[ -\frac{GM}{R + h} = -5.4 \times 10^7 \]
- From the acceleration due to gravity equation:
\[ \frac{GM}{(R + h)^2} = 6.0 \]

4. **Dividing the two equations**:
- Dividing the gravitational potential equation by the acceleration due to gravity equation:
\[ \frac{-\frac{GM}{R + h}}{\frac{GM}{(R + h)^2}} = \frac{-5.4 \times 10^7}{6.0} \]
- Simplifying gives:
\[ (R + h) = \frac{5.4 \times 10^7}{6.0} \]

5. **Calculating \( R + h \)**:
- Calculate the right-hand side:
\[ R + h = \frac{5.4 \times 10^7}{6.0} = 9.0 \times 10^6 \, \text{m} = 9000 \, \text{km} \]

6. **Finding the height \( h \)**:
- Now, we can find \( h \):
\[ h = (R + h) - R = 9000 \, \text{km} - 6400 \, \text{km} = 2600 \, \text{km} \]

### Final Answer:
The height \( h \) from the surface of the Earth is \( 2600 \, \text{km} \).