How much energy will be necessary for making a body of 500 kg escape from the earth
`[g=9.8 ms^(2)," radius of earth "=6.4xx10^(6)m]`

Options

A. About `9.8xx10^(6) J`
B. About `6.4xx10^(8) J`
C. About `3.1xx10^(10) J`
D. About `27.4xx10^(22) J`

Answer

To determine how much energy is necessary for a body of 500 kg to escape from the Earth, we need to calculate the gravitational potential energy at the surface of the Earth. The formula for gravitational potential energy (U) is given by:

\[ U = -\frac{GMm}{r} \]

Where:
- \( G \) is the universal gravitational constant, approximately \( 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)
- \( M \) is the mass of the Earth, approximately \( 5.97 \times 10^{24} \, \text{kg} \)
- \( m \) is the mass of the object (in this case, 500 kg)
- \( r \) is the radius of the Earth, approximately \( 6.4 \times 10^6 \, \text{m} \)

However, we can simplify this calculation using the acceleration due to gravity \( g \) at the Earth's surface, which is given as \( 9.8 \, \text{m/s}^2 \). The potential energy can also be expressed as:

\[ U = -mgh \]

Where \( h \) is the height (which is equal to the radius of the Earth when considering escape from the surface).

### Step-by-Step Solution:

1. **Identify the values**:
- Mass of the body, \( m = 500 \, \text{kg} \)
- Acceleration due to gravity, \( g = 9.8 \, \text{m/s}^2 \)
- Radius of the Earth, \( r = 6.4 \times 10^6 \, \text{m} \)

2. **Calculate the gravitational potential energy**:
\[ U = -mgh = -500 \times 9.8 \times (6.4 \times 10^6) \]

3. **Calculate the value**:
\[ U = -500 \times 9.8 \times 6.4 \times 10^6 \]
\[ U = -500 \times 9.8 \times 6.4 = -3.136 \times 10^{10} \, \text{J} \]

4. **Interpret the result**:
The energy required to escape the gravitational field of the Earth is equal to the absolute value of the gravitational potential energy calculated, which is approximately:
\[ E = 3.136 \times 10^{10} \, \text{J} \]

### Final Answer:
The energy necessary for making a body of 500 kg escape from the Earth is approximately \( 3.136 \times 10^{10} \, \text{J} \).