How much energy will be necessary for making a body of 500 kg escape from the earth ?[g = 9.8m / s², radius of earth =6.4 x 10⁶ m]

To find the energy necessary for a body to escape from the Earth's gravitational pull, we need to calculate its escape energy (binding energy) at the Earth's surface.

The gravitational potential energy ($U$) of a body of mass $m$ on the surface of the Earth is given by the formula:

$U=-\frac{GMm}{R}$

where:
- $G$ is the universal gravitational constant,
- $M$ is the mass of the Earth,
- $R$ is the radius of the Earth.

To completely escape the Earth's gravitational field, the body must reach infinity with at least zero total energy. Therefore, the minimum escape energy ($E$) required is equal in magnitude to the binding energy:

$E=\frac{GMm}{R}$

We know that the acceleration due to gravity ($g$) at the surface of the Earth is related to $G$ and $M$ by:

$g=\frac{GM}{R^2}$

Rearranging this gives:

$GM=g{R}^2$

Substituting $GM$ back into our escape energy equation yields:

$E=\frac{\left(g{R}^2\right)m}{R}$

Simplifying this formula, we get:

$E=mgR$

Now, we substitute the given values into the equation:
- Mass of the body, $m=500\text{kg}$
- Acceleration due to gravity, $g=9.8{\text{m/s}}^2$
- Radius of the Earth, $R=6.4\times {10}^6\text{m}$

Calculating the escape energy:

$E=500\times 9.8\times 6.4\times {10}^6$

$E=4900\times 6.4\times {10}^6$

$E=31360\times {10}^6\text{J}$

$E=3.136\times {10}^{10}\text{J}$

Thus, the energy required is approximately $3.1\times {10}^{10}\text{J}$.