At what height above the surface of earth , acceleration due to gravity will be (i) `4%` , (ii) `50%` of its value on the surface of the earth ? Given , radius of the earth = 6400 km .

Answer

To find the height above the Earth's surface where the acceleration due to gravity is a certain percentage of its value at the surface, we can use the formula for gravitational acceleration at a distance from the center of the Earth:

\[ g' = \frac{g_m}{(r + h)^2} \]

where:
- \( g' \) is the acceleration due to gravity at height \( h \),
- \( g_m \) is the acceleration due to gravity at the surface of the Earth,
- \( r \) is the radius of the Earth,
- \( h \) is the height above the Earth's surface.

The value of \( g_m \) at the surface of the Earth is approximately \( 9.81 \, \text{m/s}^2 \).

### Part (i): When \( g' = 4\% \) of \( g_m \)

1. **Set up the equation:**
\[ g' = 0.04 g_m \]

2. **Substituting the formula:**
\[ 0.04 g_m = \frac{g_m}{(r + h)^2} \]

3. **Cancel \( g_m \) from both sides:**
\[ 0.04 = \frac{1}{(r + h)^2} \]

4. **Taking the reciprocal:**
\[ (r + h)^2 = \frac{1}{0.04} = 25 \]

5. **Taking the square root:**
\[ r + h = 5 \]

6. **Substituting the radius of the Earth (in km):**
\[ r = 6400 \, \text{km} \]
\[ 6400 + h = 5 \implies h = 5 - 6400 \]
\[ h = 5 - 6400 = -6395 \, \text{km} \]

Since height cannot be negative, we realize we need to use the correct interpretation. The height \( h \) should be calculated as \( h = 5r - r = 4r \).

7. **Calculating height:**
\[ h = 4 \times 6400 \, \text{km} = 25600 \, \text{km} \]

### Part (ii): When \( g' = 50\% \) of \( g_m \)

1. **Set up the equation:**
\[ g' = 0.5 g_m \]

2. **Substituting the formula:**
\[ 0.5 g_m = \frac{g_m}{(r + h)^2} \]

3. **Cancel \( g_m \) from both sides:**
\[ 0.5 = \frac{1}{(r + h)^2} \]

4. **Taking the reciprocal:**
\[ (r + h)^2 = 2 \]

5. **Taking the square root:**
\[ r + h = \sqrt{2} \]

6. **Substituting the radius of the Earth:**
\[ 6400 + h = \sqrt{2} \implies h = \sqrt{2} - 6400 \]

7. **Calculating height:**
\[ h = \sqrt{2} \times 6400 - 6400 = (1.414 - 1) \times 6400 \approx 0.414 \times 6400 \approx 2649.6 \, \text{km} \]

### Final Answers:
- For \( g' = 4\% \) of \( g_m \), the height \( h \approx 25600 \, \text{km} \).
- For \( g' = 50\% \) of \( g_m \), the height \( h \approx 2649.6 \, \text{km} \).