How much energy will be needed for a body of mass 100kg to escape from the earth- `(G = 10M//S^(2)` and radius of earth = 6.4 xx 10^(6) m)`

Options

A. `6.4 xx 10^(9)` joule
B. `8 xx 10^(6)` jule
C. `4 xx 10^(16)` jule
D. zero

Answer

To find the energy needed for a body of mass 100 kg to escape from the Earth, we can use the concept of gravitational potential energy and escape velocity. The energy required for a body to escape the gravitational pull of the Earth can be calculated using the formula:

\[ E = mgh \]

However, in this case, we can also use the escape velocity formula to determine the kinetic energy required for escape.

### Step-by-Step Solution:

1. **Identify the given values:**
- Mass of the body, \( m = 100 \, \text{kg} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)
- Radius of the Earth, \( r = 6.4 \times 10^6 \, \text{m} \)

2. **Calculate the escape velocity (v):**
The escape velocity from the surface of the Earth is given by the formula:
\[ v = \sqrt{2gr} \]
Substituting the values:
\[ v = \sqrt{2 \times 10 \, \text{m/s}^2 \times 6.4 \times 10^6 \, \text{m}} \]

3. **Calculate the value inside the square root:**
\[ 2gr = 2 \times 10 \times 6.4 \times 10^6 = 128 \times 10^6 = 1.28 \times 10^8 \]

4. **Calculate the escape velocity:**
\[ v = \sqrt{1.28 \times 10^8} = 1.13 \times 10^4 \, \text{m/s} \]

5. **Calculate the kinetic energy (KE) required for escape:**
The kinetic energy required to reach this escape velocity is given by:
\[ KE = \frac{1}{2} mv^2 \]
Substituting the values:
\[ KE = \frac{1}{2} \times 100 \, \text{kg} \times (1.13 \times 10^4 \, \text{m/s})^2 \]

6. **Calculate \( (1.13 \times 10^4)^2 \):**
\[ (1.13 \times 10^4)^2 = 1.2769 \times 10^8 \]

7. **Calculate the kinetic energy:**
\[ KE = \frac{1}{2} \times 100 \times 1.2769 \times 10^8 = 6.3845 \times 10^9 \, \text{J} \]

8. **Final result:**
Rounding off, the energy needed for the body to escape from the Earth is approximately:
\[ KE \approx 6.4 \times 10^9 \, \text{J} \]

### Conclusion:
The energy required for a body of mass 100 kg to escape from the Earth is approximately \( 6.4 \times 10^9 \, \text{J} \).

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