The altitude at which the weight of a body is only 64% of its weight on the surface of the earth is (Radius of the earth is 6400 km)

Options

A. 1600m
B. 16m
C. 160km
D. 1600km

Answer

To find the altitude at which the weight of a body is only 64% of its weight on the surface of the Earth, we can follow these steps:

### Step 1: Understand the relationship between weight and gravitational acceleration
The weight of a body at the surface of the Earth is given by:
\[ W = m \cdot g \]
where \( m \) is the mass of the body and \( g \) is the acceleration due to gravity at the surface of the Earth.

At a height \( h \) above the surface, the weight \( W_h \) is given by:
\[ W_h = m \cdot g_h \]
where \( g_h \) is the acceleration due to gravity at height \( h \).

### Step 2: Use the formula for gravitational acceleration at height
The gravitational acceleration at height \( h \) can be expressed as:
\[ g_h = \frac{g \cdot R^2}{(R + h)^2} \]
where \( R \) is the radius of the Earth.

### Step 3: Set up the equation based on the given condition
According to the problem, the weight at height \( h \) is 64% of the weight at the surface:
\[ W_h = 0.64 \cdot W \]
Substituting the expressions for weight, we get:
\[ m \cdot g_h = 0.64 \cdot (m \cdot g) \]
This simplifies to:
\[ g_h = 0.64 \cdot g \]

### Step 4: Substitute the expression for \( g_h \)
Now substituting for \( g_h \):
\[ \frac{g \cdot R^2}{(R + h)^2} = 0.64 \cdot g \]
We can cancel \( g \) from both sides (assuming \( g \neq 0 \)):
\[ \frac{R^2}{(R + h)^2} = 0.64 \]

### Step 5: Solve for \( R + h \)
Taking the square root of both sides:
\[ \frac{R}{R + h} = 0.8 \]
Cross-multiplying gives:
\[ R = 0.8(R + h) \]
Expanding the right side:
\[ R = 0.8R + 0.8h \]
Rearranging gives:
\[ R - 0.8R = 0.8h \]
\[ 0.2R = 0.8h \]
Dividing both sides by 0.8:
\[ h = \frac{0.2R}{0.8} = \frac{R}{4} \]

### Step 6: Substitute the value of \( R \)
Given that the radius of the Earth \( R = 6400 \) km:
\[ h = \frac{6400}{4} = 1600 \text{ km} \]

### Conclusion
The altitude at which the weight of a body is only 64% of its weight on the surface of the Earth is:
\[ \boxed{1600 \text{ km}} \]

---