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

Answer

To find the height from the surface of the Earth where the gravitational potential \( V \) is \( -5.4 \times 10^7 \, \text{J kg}^{-1} \) and the value of \( g \) is \( 6.0 \, \text{m s}^{-2} \), we can use the following formulas:

1. The gravitational potential \( V \) at a height \( h \) from the surface of the Earth is given by:
\[ V = -\frac{GM}{R + h} \]
where \( G \) is the gravitational constant (\( 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)), \( M \) is the mass of the Earth (\( 5.97 \times 10^{24} \, \text{kg} \)), and \( R \) is the radius of the Earth.

2. The value of \( g \) at height \( h \) is given by:
\[ g = \frac{GM}{(R + h)^2} \]

Given:
- \( V = -5.4 \times 10^7 \, \text{J kg}^{-1} \)
- \( g = 6.0 \, \text{m s}^{-2} \)
- \( R = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} \)

### Step 1: Rearranging the gravitational potential formula

From the gravitational potential formula:
\[ -5.4 \times 10^7 = -\frac{GM}{R + h} \]
This can be rearranged to find \( R + h \):
\[ R + h = \frac{GM}{5.4 \times 10^7} \]

### Step 2: Finding \( GM \)

Using the value of \( g \):
\[ g = \frac{GM}{(R + h)^2} \]
Rearranging gives:
\[ GM = g(R + h)^2 \]

### Step 3: Equating the two expressions for \( GM \)

Set the two expressions for \( GM \) equal:
\[ \frac{GM}{5.4 \times 10^7} = g(R + h)^2 \]
Substituting \( g = 6.0 \):
\[ \frac{GM}{5.4 \times 10^7} = 6.0(R + h)^2 \]

### Step 4: Substitute \( GM \) from the second equation into the first

Now substituting \( GM \) from the second equation into the first:
\[ \frac{6.0(R + h)^2}{5.4 \times 10^7} = 6.0(R + h)^2 \]
This simplifies to:
\[ \frac{1}{5.4 \times 10^7} = 1 \]

### Step 5: Solving for \( R + h \)

Now we can solve for \( R + h \):
\[ R + h = \sqrt{\frac{GM}{6.0}} = \sqrt{\frac{g(R + h)^2}{6.0}} \]

### Step 6: Finding \( h \)

Now we can find \( h \):
\[ h = (R + h) - R \]

### Step 7: Substitute \( R \) and solve for \( h \)

Substituting \( R = 6400 \times 10^3 \, \text{m} \) and solving for \( h \):
\[ h = \frac{GM}{6.0} - R \]

### Final Calculation

Using the values of \( G \) and \( M \) to find \( GM \):
\[ GM = 6.67 \times 10^{-11} \times 5.97 \times 10^{24} \]
Calculate \( GM \) and then substitute back to find \( h \).

### Conclusion

After performing the calculations, we can find the height \( h \) above the surface of the Earth where the gravitational potential and \( g \) are as given.