Question

The energy required to project a body of mass \(2\,\text{kg}\) so that it escapes from the gravitational influence of the earth is \[ (\text{Acceleration due to gravity on the surface of the earth}=10\,\text{m s}^{-2}\text{ and the radius of the earth}=6400\,\text{km}) \]

• A. \(96\times10^6\,\text{J}\)
• B. \(32\times10^6\,\text{J}\)
• C. \(128\times10^6\,\text{J}\)
• D. \(64\times10^6\,\text{J}\)

Hint:

Remember, \[ \boxed{ E_{\text{escape}}=mgR=\frac{GMm}{R}. } \] Also, \[ \boxed{ v_e=\sqrt{2gR}. } \]

Answer 1

The Correct Option is C

Step 1: Recall the expression for escape energy. The minimum energy required for a body to escape from the Earth's gravitational field is \[ E=\frac{GMm}{R}. \] Using \[ g=\frac{GM}{R^2}, \] we obtain \[ E=mgR. \]

Step 2: Substitute the given values. Given, \[ m=2\,\text{kg}, \] \[ g=10\,\text{m s}^{-2}, \] and \[ R=6400\,\text{km} =6.4\times10^6\,\text{m}. \] Hence, \[ E = 2\times10\times6.4\times10^6. \] \[ E = 128\times10^6\,\text{J}. \] Therefore, \[ \boxed{E=128\times10^6\,\text{J}.} \] Hence, the correct option is \(\boxed{(C)}\).