All Problems Universal Gravitation
Problem 1.227
An artificial satellite of the Moon revolves in a circular orbit whose radius exceeds the radius of the Moon \(\eta\) times. In the process of motion the satellite experiences a slight resistance due to cosmic dust. Assuming the resistance force to depend on the velocity of the satellite as \(F=\alpha v^{2},\) where \(\alpha\) is a constant, find how long the satellite will stay in orbit until it falls onto the Moon's surface.
1.227. The decrease in the total energy \(E\) of the satellite over the time interval \(d t\) is equal to \(-d E=F v d t\). Representing \(E\) and \(v\) as functions of the distance \(r\) between the satellite and the centre of the Moon, we can reduce this equation to the form convenient for integration. Finally, we get \(\tau \approx(\sqrt{\eta}-1) m / \alpha \sqrt{g R}\)