Question

The energy required to take a body from the surface of the earth to a height equal to the radius of the earth is \( W \). The energy required to take this body from the surface of the earth to a height equal to twice the radius of the earth is:

• A. \(\frac{W}{3}\)
• B. \(\frac{2W}{3}\)
• C. \(W\)
• D. \(\frac{4W}{3}\)

Hint:

The gravitational potential energy change follows an inverse relationship with distance, making energy calculations crucial in celestial mechanics.

Answer 1

The Correct Option is D

The energy required to move a body from the surface of the Earth to a height \( h \) is given by: \[ W = GMm \left(\frac{1}{R} - \frac{1}{R+h}\right) \] For height \( h = R \): \[ W_1 = GMm \left(\frac{1}{R} - \frac{1}{2R}\right) = \frac{GMm}{2R} = W \] For height \( h = 2R \): \[ W_2 = GMm \left(\frac{1}{R} - \frac{1}{3R}\right) = \frac{2GMm}{3R} = \frac{4W}{3} \] Thus, the required energy is \( \frac{4W}{3} \).