How much energy should be supplied to a body of mass 500kg, so that it can escape from the gravitational pull of the earth? `[g=10 m//s^(2)` and `R=6400 km]`

Options

A. `6.4xx10^(10)J`
B. `3.2xx10^(10)J`
C. `6.4xx10^(8)J`
D. `3.2xx10^(6)J`

Explanation

To determine how much energy should be supplied to a body of mass 500 kg so that it can escape from the gravitational pull of the Earth, we can follow these steps:

### Step 1: Understand Escape Velocity
The escape velocity (\( V_e \)) is the minimum velocity required for an object to break free from the gravitational attraction of a celestial body without any further propulsion. The formula for escape velocity is given by:

\[ V_e = \sqrt{\frac{2GM_e}{R_e}} \]

where:
- \( G \) is the universal gravitational constant (\( 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)),
- \( M_e \) is the mass of the Earth (\( 5.97 \times 10^{24} \, \text{kg} \)),
- \( R_e \) is the radius of the Earth.

### Step 2: Use the Given Values
We can simplify the escape velocity formula using the acceleration due to gravity (\( g \)):

\[ V_e = \sqrt{2gR_e} \]

Given:
- \( g = 10 \, \text{m/s}^2 \)
- \( R_e = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} = 6.4 \times 10^6 \, \text{m} \)

### Step 3: Calculate Escape Velocity
Substituting the values into the escape velocity formula:

\[ V_e = \sqrt{2 \times 10 \, \text{m/s}^2 \times 6.4 \times 10^6 \, \text{m}} \]

Calculating this:

\[ V_e = \sqrt{128 \times 10^6} = \sqrt{1.28 \times 10^8} \approx 11,313.7 \, \text{m/s} \]

### Step 4: Calculate Kinetic Energy
The kinetic energy (\( KE \)) required to reach this escape velocity is given by:

\[ KE = \frac{1}{2} m V_e^2 \]

Substituting \( m = 500 \, \text{kg} \) and \( V_e \):

\[ KE = \frac{1}{2} \times 500 \, \text{kg} \times (11,313.7 \, \text{m/s})^2 \]

Calculating \( V_e^2 \):

\[ V_e^2 \approx 1.28 \times 10^8 \, \text{m}^2/\text{s}^2 \]

Now substituting this back into the kinetic energy formula:

\[ KE = \frac{1}{2} \times 500 \times 1.28 \times 10^8 \]
\[ KE = 250 \times 1.28 \times 10^8 = 3.2 \times 10^{10} \, \text{J} \]

### Step 5: Final Answer
Thus, the energy that should be supplied to the body to escape from the gravitational pull of the Earth is:

\[ \boxed{3.2 \times 10^{10} \, \text{J}} \]