Find the percentage decrease in the acceleration due to gravity when a body is take from the surface of earth to a height of 32 km from its surface. `["Take" R_(e )=6.4xx10^(6)m]`

Explanation

To find the percentage decrease in the acceleration due to gravity when a body is taken from the surface of the Earth to a height of 32 km, we can follow these steps:

### Step 1: Understand the formula for acceleration due to gravity at height
The acceleration due to gravity at a height \( h \) from the surface of the Earth can be approximated using the formula:
\[ g_h = g \left(1 - \frac{2h}{R_e}\right) \]
where:
- \( g_h \) is the acceleration due to gravity at height \( h \),
- \( g \) is the acceleration due to gravity at the surface of the Earth (approximately \( 9.8 \, \text{m/s}^2 \)),
- \( R_e \) is the radius of the Earth (given as \( 6.4 \times 10^6 \, \text{m} \)),
- \( h \) is the height above the Earth's surface (given as \( 32 \, \text{km} = 32,000 \, \text{m} \)).

### Step 2: Substitute the values into the formula
Substituting the values into the formula:
\[ g_h = 9.8 \left(1 - \frac{2 \times 32000}{6.4 \times 10^6}\right) \]

### Step 3: Calculate the fraction
Calculating the fraction:
\[ \frac{2 \times 32000}{6.4 \times 10^6} = \frac{64000}{6400000} = \frac{1}{100} = 0.01 \]

### Step 4: Substitute back into the formula
Now substituting this back into the equation for \( g_h \):
\[ g_h = 9.8 \left(1 - 0.01\right) = 9.8 \times 0.99 = 9.702 \, \text{m/s}^2 \]

### Step 5: Calculate the percentage decrease
The percentage decrease in acceleration due to gravity can be calculated using the formula:
\[ \text{Percentage decrease} = \frac{g - g_h}{g} \times 100 \]
Substituting the values:
\[ \text{Percentage decrease} = \frac{9.8 - 9.702}{9.8} \times 100 \]

### Step 6: Calculate the difference
Calculating the difference:
\[ 9.8 - 9.702 = 0.098 \]

### Step 7: Final calculation
Now substituting this difference back into the percentage decrease formula:
\[ \text{Percentage decrease} = \frac{0.098}{9.8} \times 100 \approx 1\% \]

Thus, the percentage decrease in the acceleration due to gravity when a body is taken from the surface of the Earth to a height of 32 km is approximately **1%**.

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