Problem 63 - A Complete Resource Book in Physics for JEE Main

Question

The kinetic energy needed to project a body of mass \(m\) from the earth's surface (radius \(R\) ) to infinity is (A) \(\frac{m g R}{2}\) (B) \(2 m g R\) (C) \(m g R\) (D) \(\frac{m g R}{4}\)

Step-by-Step Solution

Answer

Based on the solution steps above, the kinetic energy needed to project a body of mass \(m\) from the earth's surface (radius \(R\)) to infinity is (C) \(mgR\).

Step 1: Identify Given Information

In this exercise, we are given the mass (\(m\)) of a body that we are projecting from the earth's surface to infinity. We need to calculate the kinetic energy needed to accomplish this.

Step 2: Understand the Concept of Energy Conservation

According to the principle of conservation of energy, the total mechanical energy at the surface of the earth (i.e., kinetic energy \(KE_{surface}\) + gravitational potential energy at the surface \(PE_{surface}\) ) should be equal to the total mechanical energy at infinity, which is just the gravitational potential energy at infinity, \(PE_{infinity}\). Since at infinity, the kinetic energy of the body is 0 and potential energy is -0 i.e., total energy is 0, our equation of energy conservation becomes: \(PE_{surface} + KE_{surface} = 0\).

Step 3: Calculate Potential Energy at the Surface

The gravitational potential energy at the surface of the earth is given by: \(PE_{surface} = - \frac{G m M}{R}\) , where M is the mass of the earth, \(G\) is the gravitational constant, and \(R\) is the radius of the earth. We can substitute the gravitational force expression \( F = mg = G m M / R^2 \) in our equation. This gives us: \(PE_{surface} = - mgR\).

Step 4: Calculate Kinetic Energy at the Surface

Substituting the potential energy \(PE_{surface}\) we calculated into the equation from Step 2, we get the kinetic energy needed: \(PE_{surface} + KE_{surface} = 0 ⟹ KE_{surface} = - PE_{surface} = mgR\).