If `g` on the surface of the earth is `9.8 m//s^2`, its value at a height of `6400km` is (Radius of the earth `=6400km`)

Options

A. `4.9 ms^(-2)`
B. `9.8 ms^(-2)`
C. `2.45ms^(-2)`
D. `19.6ms^(-2)`

Explanation

To find the value of gravitational acceleration \( g' \) at a height of \( 6400 \, \text{km} \) above the Earth's surface, we can use the formula for gravitational acceleration at a height \( h \):

\[ g' = \frac{g}{(1 + \frac{h}{R})^2} \]

Where:
- \( g \) is the gravitational acceleration at the surface of the Earth, which is \( 9.8 \, \text{m/s}^2 \).
- \( h \) is the height above the Earth's surface, which is \( 6400 \, \text{km} \).
- \( R \) is the radius of the Earth, which is also \( 6400 \, \text{km} \).

### Step 1: Convert height and radius to the same units
Since both \( h \) and \( R \) are given in kilometers, we can keep them in kilometers for our calculations.

### Step 2: Substitute values into the formula
Substituting \( h = 6400 \, \text{km} \) and \( R = 6400 \, \text{km} \) into the formula:

\[ g' = \frac{9.8}{(1 + \frac{6400}{6400})^2} \]

### Step 3: Simplify the expression inside the parentheses
Calculating the term inside the parentheses:

\[ 1 + \frac{6400}{6400} = 1 + 1 = 2 \]

### Step 4: Substitute back into the formula
Now substitute back into the equation:

\[ g' = \frac{9.8}{(2)^2} = \frac{9.8}{4} \]

### Step 5: Calculate the final value
Now, calculate \( \frac{9.8}{4} \):

\[ g' = 2.45 \, \text{m/s}^2 \]

### Final Answer
The value of gravitational acceleration at a height of \( 6400 \, \text{km} \) above the Earth's surface is:

\[ g' = 2.45 \, \text{m/s}^2 \]