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Universal Gravitation

Problem 1.227

An artificial satellite of the Moon revolves in a circular orbit whose radius exceeds the radius of the Moon η\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=αv2,F=\alpha v^{2}, where α\alpha is a constant, find how long the satellite will stay in orbit until it falls onto the Moon's surface.

Reveal Answer
1.227. The decrease in the total energy EE of the satellite over the time interval dtd t is equal to −dE=Fvdt-d E=F v d t. Representing EE and vv as functions of the distance rr between the satellite and the centre of the Moon, we can reduce this equation to the form convenient for integration. Finally, we get τ≈(η−1)m/αgR\tau \approx(\sqrt{\eta}-1) m / \alpha \sqrt{g R}
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