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Chapter 7: Problem 63

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) \(\frac{m g R}{4}\)

Short Answer

Expert verified
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 by step solution

01

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.
02

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: \(PE_{surface} + KE_{surface} = 0\).
03

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: \(PE_{surface} = - mgR\).
04

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: \(PE_{surface} + KE_{surface} = 0 鉄� KE_{surface} = - PE_{surface} = mgR\).

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